Renato Caminha
email: caminha@unistra.fr

Juacaba Neto

Pascal Merindol
email: merindol@unistra.fr

Antoine Gallais
email: antoine.gallais@uphf.fr

Fabrice Theoleyre
email: theoleyre@unistra.fr

Scalability of LPWAN for Smart City Applications

Keywords: Smart Cities, Intelligent Transportation Systems, LPWAN, LoRa, scalability

Smart Cities rely on a broad variety of services generating an heterogeneous traffic load, and having diverse privacy and performance requirements. In this paper, we focus on mobility services for an Intelligent Transportation System (ITS), where several applications inter-operate by sharing information.

In particular, we consider three main services: ride-hailing, smart parkings and traffic regulation. To investigate the load generated by these mobility services, we rely on two real datasets. Our goal is to study whether a given technology can scale with realistic conditions related to the envisioned smart mobility scenario. In particular, we analyze here how many LoRa gateways are necessary and how they should be deployed to support these mobility services. We highlight the bottlenecks occurring in most dense areas of the network and argue that heterogeneous deployments can efficiently handle such hot spots.

I. INTRODUCTION

The plethora of smart city services highlights the various benefits that Internet of Things (IoT) can bring to the population, companies and local authorities [START_REF] Lin | A Survey on Internet of Things: Architecture, Enabling Technologies, Security and Privacy, and Applications[END_REF]. A collection of sensors is typically disseminated in a smart city, and monitors a large set of environmental parameters such as traffic speed, air quality, weather, noise, etc. These data streams help to make the city more reactive and resilient. One of the most promising services is about road traffic detection sensors to re-route vehicles through less congested roads and indicate them best spots to park.

Nowadays, smart city services are deployed as specific applications, that emerge independently like as many proofs of concepts (e.g., charging stations for electric vehicles, waste management) for the future smart cities. But overall, smart cities expect that the different applications cohabit with each other exchanging data. Indeed, developing a silo architecture is inefficient: applications are isolated and have to deploy their own sensors even if they process the same kind of data. Besides, such isolation has a cost [START_REF] Marquez | How Should I Slice My Network? A Multi-Service Empirical Evaluation of Resource Sharing Efficiency[END_REF]: the same radio resources have to be shared among a larger number of flows, which is particularly prejudicial in the ISM band.

While most deployments have demonstrated the interest of connecting a myriad of sensors and actuators, we aim to investigate the scalability of Low Power Wide Area Networks (LPWAN) to handle such services. It is absolutely necessary to avoid the tragedy of the commons [START_REF] Feeney | Avoiding an IoT 'Tragedy of the Commons[END_REF]. Indeed, while each object has only a few packets to transmit infrequently, the competition for the medium access involves many devices.

LPWAN have already highlighted their limits when the load (aka bit flux) increases [START_REF] Ghena | Challenge: Unlicensed LPWANs Are Not Yet the Path to Ubiquitous Connectivity[END_REF], limiting the amount of traffic that an application can generate.

In this paper, we investigate the scalability of LPWAN solutions, and more specifically LoRa, a leading technology for LPWAN, when considering a challenging smart mobility solution. We consider both mobile devices, in particular to monitor the actual speed through many road segments, and static devices used in smart parking. We also consider a ride hailing service co-existing in the same city. The contributions of this paper are as follows:

1) first, we define a global scenario for Intelligent Transport System (ITS): distinct applications share data to offer a globally efficient transportation system for smart cities; 2) then, we take advantage of two real and large cities datasets for our global system. That way, we can generate packets that mimics a realistic situation, with all the heterogeneity (e.g., downtown vs. suburbs) and temporality of large scale ITS. The traffic model we derive from these datasets is useful to assess the communication performances in real smart cities; 3) finally, to fully exploit our construct, we rely on a LoRa model inferring the Packet Error Rate (PER) of an arbitrary deployment of LoRa gateways; we thus estimate the scalability of such LPWAN solutions in realistic deployments. We highlighted the presence of hot spots, where the density of required LoRa gateways can be quite high to decently deal with some traffic peaks.

II. RELATED WORK

We detail here a few applications defined for smart cities, and how to characterize them. Then, we discuss the existing LPWAN scalability evaluations.

A. Service characterization

Realistic service definitions are crucial in order to assess suitability of proposals. It is common to characterize a service by the amount of messages generated by device and/or geographical area [START_REF] Union | Guidelines for evaluation of radio interface technologies for IMT-2020[END_REF]. Fortunately, more and more open data projects have emerged [START_REF] Ahlgren | Internet of Things for Smart Cities: Interoperability and Open Data[END_REF] and allow us to design specific scenarios. Zebranet [START_REF] Zhang | Hardware Design Experiences in ZebraNet[END_REF] represents an early example of such projects which has experimentally investigated the accuracy of Wireless Sensor Networks for wildlife monitoring. The Zebranet dataset has then been widely used, to assess the performance of protocols and algorithms. In Smart Cities, Kong et al. [START_REF] Kong | Mobility Dataset Generation for Vehicular Social Networks Based on Floating Car Data[END_REF] exploit a floating car dataset in New York City (NYC) to simulate a Vehicular Social Network.

However and unfortunately, most of these deployments focus only on a single service and, as such, remain silo proofs of concepts. Mixing different services having distinct requirements is a more interesting challenge and certainly represents most common scenarios to come.

B. LPWAN and Scalability

Low Power Wide Range technologies enable smart cities by providing the ability to connect a large number of spread devices required by these services. In particular, LoRa [START_REF] Sornin | LoRaWAN Specification[END_REF] is capable of reaching several kilometers of range and thousands of low power devices per base station. The deployment then looks easier: a few antennas, aka. LoRa gateways, are sufficient in theory to provide a large scale connectivity.

LoRa is based on a Chirp Spread-Spectrum (CSS) modulation technique, that uses pulses to encode information. More precisely, a pair of transmitter/receiver has to select a spreading code, also designated as a Spreading Factor (SF) in LoRA. By selecting the right SF, each device can choose the highest bit rate adapted to its signal strength. More importantly, the SFs are mutually orthogonal, to multiplex the transmissions. That is, transmissions on different SFs never collide and SFs have different ranges, air times and bit rates. However, handling a very large number of devices (even generating only a few packets each) with few antennas is challenging [START_REF] Ghena | Challenge: Unlicensed LPWANs Are Not Yet the Path to Ubiquitous Connectivity[END_REF]. Collisions are frequent and network capacity may be rapidly reached.

Characterizing the services is not enough: one needs to estimate the reliability provided by the wireless infrastructure to be certain that it matches the requirements. Bankov et al. [START_REF] Bankov | Mathematical model of LoRaWAN channel access with capture effect[END_REF] provide a model that computes the PER based on the signal strength of all the devices, the amount of packets they generate, and the allocation of SF to each device. The capture effect can significantly impact the reliability of communications in large geographical areas [START_REF] Fernandes | On the Real Capacity of LoRa Networks: The Impact of Non-Destructive Communications[END_REF], it should also be modeled to provide more realistic results.

III. INTELLIGENT TRANSPORTATION SERVICES

Intelligent Transportation represents a key enabler for smart cities. We consider here the following mobility services:

Ride hailing a platform connects clients with vehicle drivers (aka. on-demand taxi). In a smart city, this service could be provided locally. Smart parking where sensors monitor occupied parking spots. Vehicles can then be guided to find an available parking spot, and are billed by the system on departure. Traffic monitoring where the ITS measures the average speed for each road segment. This information helps for traffic regulation (e.g., vehicle routing problem). We rely on a slightly extended version of SQL to explicitly define the queries (i.e., the arrows in Fig. 1) that trigger the streams of data between entities (in the inverse direction of the queries). Furthermore, we map relational to IoT data by having data sources, samples and attributes acting as tables, rows and columns, respectively. We then simply add the following keywords in our query language extending SQL to fully describe the IoT streams: COMPUTE indicates a list of computations to apply to the data in sequence. EVENT describes condition(s) that may activate/deactivate a stream (anything, from local sensors to remote messages). EVERY indicates when new data must be generated to the stream. Events or constant time intervals may apply. UNTIL indicates the condition to stop the stream. Analogous to EVERY, events or specific duration may apply. Without this keyword, the query is assumed to hold indefinitely.

B. Ride Hailing

Recently, the portion of taxi trips being handled by private cars has been steadily increasing. A common platform serves as a broker between passengers and drivers, so that all the requests can be fulfilled. We detail here streams 1 and 2 illustrated in Fig. 1.

1) Query 1 (ride selection for passengers): passengers request companies (i.e., brokers) for best quotes to perform a ride by specifying the time of departure, and the geographical location of departure and arrival. The reply data stream consists of a continuous query remaining active until the passenger has validated one of the proposed quotes. The data stream stops when the passenger accepts a quote. 2) Query 2 (ride selection for drivers): for the other part of the service, drivers must select their next ride. More precisely, the drivers trigger a continuous query to get unsatisfied rides when they are close to complete their current fare. We assume here that brokers implement a scoring function (ride_score) which ranks rides according to characteristics such as driver preferences, proximity, and expected payment. As soon as a driver accepts a quote, the corresponding query stops.

C. Smart Parking Service

Searching for a parking spot is particularly expensive in terms of time and fuel [START_REF] Mangiaracina | Smart parking management in a smart city: Costs and benefits[END_REF]. Thus, smart parking solutions help drivers to find a parking spot, and to reduce their waiting time. We can also make the system more efficient, with an automatic billing solution.

1) Query 3 (Smart Billing): A vehicle can be automatically billed when exiting its parking spot. We consider that a sensor is capable of detecting the occupation of one or multiple parking spots and then sends notifications whenever a vehicle enters or departs of a spot. This notification is sent to the company so that the vehicle can be charged accordingly. These notifications include what event occurred (arrival or departure), timestamp of the event, and identifications of vehicle and spot.

1 SELECT event_type, time_stamp, car_plate, parking_spot_id 2 FROM parking_parking 3 EVERY EVENT park_begin, park_end Query 3: Tracking vehicles entrances.

2) Query 4 (Guidance of cars to available parking): Drivers ask for the availability of nearby parking areas to find an available parking spot. A filter selects areas with a sufficient number of available spots (THRESHOLD_VALUE), else the available spots may be occupied by the time the driver arrives to the location. Then, results are ordered by proximity and limited to prevent verbose answers. The continuous query 4 holds from the time of departure to the time of arrival (end of trip), and the list is updated every minute.

D. Smart Road Traffic Routing Service

Traffic jams induce increasing costs in most major cities, both in terms of time, money and pollution 1 . Not to mention the frustration caused to drivers and consumers and thus the ensuing damages it may lead to.

Vehicle routing solutions help drivers to find better routes [START_REF] Falek | To Re-Route, or not to Re-Route:Impact of Real-Time Re-Routing in Urban Road Networks[END_REF] to mitigate these effects. It requires road surveillance in order to identify the level of congestion of each road segment.

1) Query 5 (Real-time Road Speeds): A smart mobility service may collect the location and speed measurements from a large collection of vehicles (e.g., bus, private cars, taxis) to profile congestion in real time. Smart cities may exploit this information to make some interpolations and infer congestion in each road segment. The real identity of each vehicle should be hidden (e.g. with a hash) for obvious privacy concerns.

1 SELECT hash(car_plate), latitude, longitude, current_speed 2 FROM consenting_vehicles 3 EVERY 3 minutes Query 5: Road congestion information.

IV. OPEN MOBILITY DATASETS

In this section, we detail how we exploit two public datasets to model the global smart mobility service we envision. We rely on this data for performance evaluation: can an ITS relying on LoRa gateways can scale in these realistic conditions?

A. Datasets Description

We exploited the two following datasets (cf. Table I). Ride hailing: we use a dataset of rides performed by the companies Uber, Juno, Lyft and Via in NYC 2 . Each sample describes the pick-up and drop-off information (i.e., timestamp and geographical location) of a trip. To respect privacy concerns, the city employs pre-defined zones and zone identifiers instead of exact geographical coordinates.

We model here mobile sensors (i.e., vehicles); Parking usage: the city of Melbourne provides measurements of parking sensors 3 . Each sample of this dataset describes individual occupations of parking spots in the city and includes timestamps of arrival and departures in addition to multiple identifiers of the parking spot. We have here static sensors with an event-based traffic.

B. Dataset Preparation

Since the service usage described by these two datasets are deployed in different cities, we need to merge both to mimic a global service running in the same city. We map Melbourne's dataset to NYC since more information is openly available for the latter. We gather a list of parking lots in NYC with the aid of the TomTom API 4 . Since we have more parking lots in Melbourne than in NYC, one-to-one mapping cannot be used. We first cluster the 4,876 parking lots in melbourne into 1,212 clusters (the number of parking lots we found in NYC) using the k-means algorithm according to geographical location. With this, each cluster represents a virtual parking lot with the data from its members. Now we can look for a parking bijection from Melbourne to NYC. Assuming that parking lots closer to Manhattan (commercial center of NYC) will have more demand, we map our virtual parking lots of Melbourne with most traffic into the parking lots of NYC which are closer to Manhattan.

To mimic vehicle displacement, we expand the NYC dataset by computing the trajectory between pick-up and drop-off locations for each trip. For the sake of simplicity, we consider here straight line trajectories under constant speeds. In our case, an approximated geographical location is sufficient to model the LoRa traffic.

C. Database-based Network Traffic Model

We infer the network traffic from the datasets (cf. Table II that summarizes the parameters used in our models):

1) Ride Hailing: we model both queries 1 and 2 by using the time and location information of taxi pick-ups and dropoffs in NYC. Each trip record emulates a query at time and location of pick-up and drop-off. We model the duration of the query with a normal distribution (see Table II).

2) Smart Parking: we model the park_begin and park_end events of queries 3 with the arrival and departure data observed in the Melbourne dataset. More specifically, for each record, at time of arrival and time of departure, one message is sent from the parking lot location.

We model the inter-query time of query 4 by assuming that there was a search for available parking every time a parking spot is taken. Since available parking can change constantly, one message is sent every minute until the car parks. For same reasons as before, we again rely on a normal distribution to model the time to find a parking spot (see Table II). 3) Smart Road Traffic Routing: in query 5, we consider that a subset of vehicles sends its speed information (e.g., a subset of vehicles is equipped). More precisely, we consider that only 10% of the dataset of taxi trips is considered by this sub-service. Messages are sent periodically along the trajectory we generated for each trip.

V. SCALABILITY ANALYSIS WITH A LORA DEPLOYMENT

We discuss here whether a wireless technology like LoRa could be used for such Smart Mobility services. We assume that LoRa will be the only means of communication.

A. LoRa Analysis Methodology

To conduct our scalability analysis, we consider the LoRa analytical model described in [START_REF] Bankov | Mathematical model of LoRaWAN channel access with capture effect[END_REF]: for a given amount of network traffic, it predicts the PER of each device associated with a LoRa gateway. This model measures collision probabilities for each SF from the number of devices tuned to it and assuming a Poisson process for packet generation 5 .

In our calculations we use the EU 863-880 MHz ISM band, with the receiver sensitivity of the Semtech SX1276 LoRa receiver. We aim here to understand how many gateways are required to ensure an efficient large-scale mobility service deployment with a reasonable PER.

We consider two different SF allocation methods: Distance-based: some devices are too far from the LoRa gateway and we must consider their signal strength when selecting a SF. Thus, we allocate to each device the lowest SF with respect to its signal strength. A lower SF means that the bitrate is higher. Airtime-based: with very dense deployments, devices are closer to the gateway and the signal strength is not a constraint anymore. In that case, we balance the load for each SF. More precisely, we assign the data rates so that all the data rates have the same cumulative air time. The devices with the largest SF consume more energy to stay awake longer, but we reduce the number of collisions.

We use the density of base stations deployed 5km appart as the threshold to use the fair airtime allocation; we indeed observed that at this point, the airtime allocation becomes a better strategy. To simplify our study, we also assume that any message can be sent within a single LoRa transmission.

We can now rely on the ITS we design along with the real datasets introduced in sections III and IV. Since traffic can greatly vary in time, we pick 200 random time intervals of one hour, and use the average number of messages per second during each of these intervals. More specifically, for each randomly picked interval, we count the number of messages sent by each device and infer the average number of messages sent per second by each device. By considering independent enough time samples, we are able to investigate the scalability of LoRa for these smart mobility services.

We consider uniform grid deployments for LoRa gateways. To assess the scalability of such LoRa deployments, we however investigate several grid sizes, according to the density. Overall, we consider each borough separately since each area does not generate the same amount of traffic.

B. Results: Adapting the Deployment Density Looks Enough

We measured the intensity of traffic per mobility service and per borough (Fig. 2). The graph presents the distribution of the number of messages for all the 200 time intervals. The violin plots are stretched out: the number of messages is obviously much larger during peak hours. Besides, if we count the sum of messages, we can see that Manhattan concentrates most of the traffic, whatever the mobility service. For instance, 65% of ride haling messages are generated in Manhattan while the Relying on this data, we can now compute the distribution of the PER independently for each borough (Fig. 3). Due to space limitations, we limit our plots to Manhattan and Staten Island (with respectively the largest and the lowest number of messages per second in NYC). While each gateway is theoretically capable of serving nodes that are up to 9km away, the traffic is too high and the network is in that case congested. For instance, we can see that two gateways are not enough to efficiently cover Staten Island. However, 4 gateways seem enough to cover the whole area, and to also decently limit the number of collisions. Similar patterns can be seen for other boroughs, except for Manhattan.

For Manhattan, the volume of packets being specially large, many collisions occur. We see that only very dense deployments of LoRa gateways are capable of providing an acceptable PER. As a direct result, devices are very close to the LoRa gateways, and the lowest SF may be used by all devices, creating congestion. Thus, we need a more balanced allocation among the different SFs, which motivates a further evaluation of the two SF allocation schemes.

Fig. 4 shows how our two allocation strategies behave under various network densities. For extremely dense deployments (i.e., one gateway every kilometer), the number of devices associated with a given gateway is quite low and collisions seldom occur: the PER is maximal. For all the other densities, the airtime-based strategy is more efficient: devices are well balances among SFs, and the system minimizes the number Finally, we look at the geographical distribution of the PER value (Fig. 5). More precisely, for each borough, we select a grid size to obtain for each device an acceptable reliability. Thus, we consider 5km spacing for Manhattan, 13 km spacing for Staten Island and 9km spacing for all other boroughs. We represent the results with a quadtree: the size of each square is selected so that each square has the same number of devices. Then, each square is colored according to the average PER obtained by the corresponding devices. As we can see, the resulting PER reaches a maximum of 4% in the center of the map: squares are smaller, denoting a huge number of devices that generate traffic. On the contrary, sparse grids are enough for Staten Island, since the traffic is very small in regards of Manhattan. If the PER has to be decreased even further, an efficient optimization scheme can be used to deploy extra base stations in the congested hop spot areas (red ones).

VI. CONCLUSION & PERSPECTIVES

We define here a global smart mobility scenario, where several services cohabit. We then emulate the traffic generated by such scenario by using two datasets, collected respectively in NYC and Melbourne. In particular, we consider both eventbased traffic from static devices, and periodical traffic from mobile ones. Finally, we analyze how a LPWAN deployment (LoRa in practice) may be sized to support these mobility services. While our scalability analysis tends to indicate that relying only on LoRa seems enough and so conceivable for an unified ITS deployment, we note that the number of LoRa gateways should be carefully tuned according to the traffic. Overall, the network capacity is reached quite fast, and may create scalability issues for large-scale scenarios.

In a future work, we plan to investigate how to refine the placement of the LoRa gateways. We propose here a simple heuristic, using a grid density adaptive to each borough. We should go further, by identifying the most accurate and favorable positions, depending on the location of hotspots. Unfortunately, the location of gateways being a continuous 2D variable, the global optimization becomes almost intractable. We may also explore how different technologies (medium vs. long range) may be complementary in such situations, handling differently static and mobile devices. Typically, smart parkings concentrate many devices in restricted areas that can result in congested hotspots.

Figure 1 Fig. 1 :

 11 Figure1illustrates our scenario, where the different applications and entities share information with each other.

  4 https://developer.tomtom.com/

Fig. 2 :

 2 Fig. 2: Message density per service and neighborhood.

Fig. 4 :Fig. 5 :

 45 Fig. 4: Impact of the SF allocation strategy in Manhattan.

  Query 1: Passenger request sent to a set of broker(s).

	1 SELECT pricing_type, cost_to($destination) as
	ride_cost
	2 FROM ride_brokers
	3 ORDER BY ride_cost DESC
	4 LIMIT 5
	5 EVERY minute
	6 UNTIL EVENT client_decision

TABLE I :

 I Datasets used in this study.

	Service	Parking usage	Private-driver hailing
	Location of sampling	Melbourne, Australia	New York City, USA
	Number of samples	37.7 million samples	234 million samples
	Sampled Period	Jan 2019 to Dec 2019	Feb 2019 to Dec 2019
	Data available Individual Parking occupations Taxi rides provided by private cars

TABLE II :

 II Service parameters.

	Q Parameter	Value
	1 Periodicity	1 minute
	Passenger decision	Normal (µ=1, σ=1) min
	2 Periodicity	5 minutes
	Driver decision	Normal (µ=5, σ=3) min
	3 Event-based	One shot query
	4 Periodicity	1 minute
	Limit	10
	Trip duration	Normal (µ=8, σ=3) min
	5 Periodicity	3 minutes
	Vehicles subset	10% of available cars

https://inrix.com/press-releases/2019-traffic-scorecard-us/

https://www1.nyc.gov/site/tlc/about/tlc-trip-record-data.page

https://data.melbourne.vic.gov.au/Transport/On-street-Car-Parking-Sensor-Data-2019/7pgd-bdf2

All formal details are given in[START_REF] Bankov | Mathematical model of LoRaWAN channel access with capture effect[END_REF].

ACKNOWLEDGMENT

This work was supported by the French National Research Agency (ANR) project Nano-Net under contract ANR-18-CE25-0003.


P Gravier

F Mas

A Barthelemy

E Boller

L Salvo

P Lhuissier
email: pierre.lhuissier@simap.grenoble-inp.fr

Pore closure in thick aluminum plate: from industrial hot rolling to individual pore observation

Keywords: Aluminum, Hot rolling, FE simulation, in situ X-ray microtomography, Pore volume evolution

Pores are often present in large aluminum ingots after casting. To ensure the mechanical reliability of the final thick plates, these pores must be closed during the forming process, hot rolling in the present case. This study aims at understanding the effects of rolling parameters on the volume evolution of pores. To do so, X-ray microtomography is used to track real casting pores during deformation. Nevertheless, thick plates are too large to enable a fine characterization of the evolution of pores during the process. The size and

shape of the samples as well as the mechanical boundary conditions must be optimized to meet imaging constraints. This paper focuses on the reproduction of complex loading paths. Multi-scale FE simulations are used to reproduce the loading conditions of thick plate rolling with uniaxial tests on samples of a few millimeters. Uniaxial tests are then characterized with tomography on a synchrotron X-ray beamline. In each sample, tens of pores are individually tracked, giving access to their volume evolution. The local loading path experienced by each pore is determined with FE simulation. It is shown that the volume evolution of a real pore correlates with the hydrostatic integration which is the integral of stress triaxiality along cumulated strain. This confirms that

Introduction

Aluminum alloys are largely used in aerospace manufacturing thanks to their low density, their good mechanical properties, and their high resistance to corrosion. Their use as thick plate in aerospace applications depends on the mastery of processing parameters. It is necessary to avoid the presence of pores in the final product and the concomitant deterioration of mechanical properties. Industrially, the soundness of the material is ensured by control tests such as dye penetrant inspection. The number of pores, their size and the distance between pores must be in a certain range. Pores in aluminum products appear during casting. They are caused by the increase of density and the decrease of hydrogen solubility at solidification. In the case of thick aluminum plate, pores are closed by plastic deformation during hot rolling. The optimization of this process is nevertheless complex. There is a large number of relevant rolling parameters (temperature, reduction per pass, roll radius, roll velocity) and their dependence on each other precludes a systematic experimental campaign at the scale of industrial rolling mill. It is essential to accurately describe the closure of pores to ensure the final soundness of the plate. [START_REF] Park | A study of void crushing in large forgings I: Bonding mechanism and estimation model for bonding efficiency[END_REF] showed that pore closure occurs in two steps. First morphology and topology of the pore evolve, along with the pore volume decrease. Then, the internal surfaces of the pore weld together during bounding.

In the following, we will only focus on the first step also called mechanical closure. [START_REF] Llanos | Improvement of central soundness in long products from a through process control of solidification and reheating and rolling[END_REF] investigated pore closure during rolling by using 2D post mortem observations. Such approach allows real pores to be investigated but does not give access to volume evolution. [START_REF] Chen | Void closure prediction in cold rolling using finite element analysis and neural network[END_REF]later Joo et al. (2014) used artificial drilled pores in order to track the pore volume evolution during rolling. They found that pore closure is influenced by the reduction of the pass, the pore size and the diameter of the rolls. However, it has been shown numerically by [START_REF] Hwang | Finite element simulations on void closure behaviour inside the sheet during sheet rolling processes[END_REF] in 2D and by [START_REF] Kb | Simulation of void defect evolvement during the forging of steel ingot[END_REF] in 3D that pore shape has a significant effect on pore closure kinetics. Since real pore surrounded by matter cannot be considered as drilled pore, [START_REF] Youssef | X-ray microtomographic characterisation of pore evolution during homogenisation and rolling of Al-6Mg[END_REF] and [START_REF] Toda | Healing behavior of preexisting hydrogen micropores in aluminum alloys during plastic deformation[END_REF] performed 3D observations of real pores. On the one hand, real complex pores were observed but they were not tracked along the rolling pass schedule. On the other hand, same pores were observed before and after hot rolling. However, in this latter case, only spherical micropores were studied whereas pores in aluminum alloys can have branched and tortuous shapes. Characterization of real complex pores during their closure will allow the understanding of their closure mechanisms and closure kinetics to improve.

Then, the current models of pore closure can be validated or improved . X-ray microtomography offers both a 3D and non destructive characterization at a spatial resolution (order of magnitude of micrometer) compatible with pore description. The non destructive aspect allows several 3D images to be captured per pore during deformation. To save time, sample size was limited to several millimeters. In this way, the whole sample and pores are imaged with sufficient resolution and a unique scan. Such a sample contains several tens of pores, providing many pore volume evolutions to investigate closure kinetics. Samples are deformed at high temperature using a cumbersome setup that requires a synchrotron X-ray source to perform interrupted in situ tests.

Consequently, there is significant difference of size between, on the one hand, industrial plate ingots that are hot rolled, which size is about 4m × 1.5m × 0.5m, and on the other hand, samples of several millimeters in which pores can be observed. It is necessary to ensure that the loading path experienced by pores during hot rolling can be reproduced with the compression of samples compatible with X-ray microtomography.

In the following, a combination of numerical and experimental approaches is presented. First, finite element simulations are used to compare the loading paths encountered during industrial thick plate rolling to the loading paths encountered during uniaxial tests of small samples with varied shapes. A dedicated experimental setup has been developed to perform high temperature in situ tests characterized by X-ray microtomography. The volume evolution of pores during their deformation are determined using 3D quantitative analysis methods.

Numerical and experimental methods

Numerical simulations

Rolling of industrial plate and a uniaxial mechanical test on a sample compatible with X-ray microtomography differ in terms of sample size and nature of loading. Finite element (FE) simulations were used to ensure that uniaxial tests are representative of the loading states reached with rolling.

Hot rolling simulations are run with LAM3 software [START_REF] Hacquin | Modelisation thermomecanique tridimensionnelle du laminage couplage bande/cylindres[END_REF]. The model accounts for thermal and mechanical phenomena involved in rolling (such as the deformation of the plate and the rolls, the thermal exchanges between the rolls and the plate, between the plate and the air and the temperature increase in the plate due to mechanical deformation). It couples finite element method to compute the elasto-viscoplastic behavior of the plate and semi-analytic computations of the deformation of the rolls. A stationary Eulerian formulation is used, thus the transient regime at both ends of the plate are not accessible. A classical rolling pass schedule for thick plates requires a progressive reduction of the plate. Several passes between the rolls are necessary to reach the desired final thickness. In the following, we will study the reproduction of the loading state of individual passes. Each pass geometry is usually described by two parameters which are the relative reduction r (Equation 1) and the L/H ratio (Equation 2), where L is the contact length between the rolls and the plate and H is the mean thickness of the plate. h 1 and h 2 are the entry and exit thicknesses respectively and R cyl is the radius of the rolls (Figure 1). 3) [START_REF] Sellars | Hot workability[END_REF].

r = h 1 -h 2 h 1 ∈ [0, 1] (1) L H = 2 R cyl (h 1 -h 2 ) h 1 + h 2 (2)
σ = 1 α ln   Z A 1 m + Z A 2 m + 1   , Z = ε exp Q RT (3)
The four parameters α, A, m and Q were determined experimentally using the method described by Zhu et al. (2018). Most of the time, the pores are studied in a matrix following a Norton power law of this type ε = A σ n . For a given temperature of 480 °C, the Norton index n of the present material is of about 6. Usually, the Norton index of aluminum alloys is between 3 and 10, which limits its effect on pore closure reported by [START_REF] Zhang | A criterion for void closure in large ingots during hot forging[END_REF]. Besides, Saby et al. (2015a) observed that the effect of the Norton exponent in this range is negligible compared to the effect of the orientation or the shape of the pore. In the simulation software, the material law consists in several stress-strain curves at different temperatures and strain rates: 10 temperatures between 380 and 580 °C and 10 strain rates between 1 × 10 -3 and 1 × 10 2 s -1 . For compression of cylinders, Brazilians and double-collars, infinitely rigid dies are used in the simulation. A coefficient of static friction of 0.4 between the die and the sample is applied. This coefficient has been chosen in order to match barreling effect that is observed experimentally during compression test. Concerning thermal conditions, the initial temperature is fixed at all sample nodes and no exchanges are considered between the dies and the air. The simulation takes into account the heat due to the mechanical work. Compression tests were simulated up to a relative reduction of 60% and tensile tests up to initiation of necking. The dimension of pores are small compared to the sample size. It is assumed that the mechanical fields are not modified by the presence of the pores. Therefore, the simulations were performed on samples free of pores. In the literature, cumulated strain (Equation 4) and stress triaxiality (Equation 5) are shown to be the two main mechanical parameters involved in pore closure. They are used in the analytical models of [START_REF] Gurson | Continuum Theory of Ductile Rupture by Void Nucleation and Growth: Part I-Yield Criteria and Flow Rules for Porous Ductile Media[END_REF] and [START_REF] Duva | Constitutive potentials for dilutely nonlinear materials[END_REF], the semi-analytical model of [START_REF] Zhang | A criterion for void closure in large ingots during hot forging[END_REF] and the phenomenological model of Saby et al. (2015a). Thus, loading states will first be compared in terms of triaxiality T X and cumulated strain ε.

ε = t 0 ε dt ε = 2 3 ε : ε ε = ∂ε ∂t (4) T X = - p σ p = - 1 3 tr σ σ = 3 2 σ : σ σ = σ + pI (5)
Where ε is the strain tensor, σ the stress tensor and p the hydrostatic pressure.

The stress state is more precisely represented considering the Lode parameter µ (Equation 6) in addition to the stress triaxiality.

µ = 2σ 2 -σ 1 -σ 3 σ 1 -σ 3 (6)
The Lode parameter influence has been mostly studied in the scope of failure of ductile metals. It has been observed by [START_REF] Danas | Influence of the Lode parameter and the stress triaxiality on the failure of elasto-plastic porous materials[END_REF] that at high triaxiality, ductile failure is controlled by triaxiality but when triaxiality decreases (in absolute value), the Lode parameter plays a significant role. [START_REF] Zhang | Numerical analysis of the influence of the Lode parameter on void growth[END_REF] demonstrated that, at low positive triaxiality, the Lode parameter has a non-negligible influence on the pore shape rather than on the pore volume. [START_REF] Chbihi | Influence of Lode angle on modelling of void closure in hot metal forming processes[END_REF] studied pore closure using FE simulations, they concluded that at low triaxiality, high values of the Lode parameter foster pore closure. More recently, [START_REF] Zhang | A porosity closure model considering stress triaxiality ratio and Lode stress parameter[END_REF] and [START_REF] Wang | A void evolution model accounting for stress triaxiality, Lode parameter and effective strain for hot metal forming[END_REF] proposed two new prediction models of pore closure that incorporate the Lode parameter.

Experimental data acquisition

A dedicated micropress has been developed ( In situ tests at high temperature are performed with a synchrotron Xray source, at the ESRF (European Synchrotron Radiation Facility, Grenoble, France) on ID19 beamline. It provides a parallel quasi-monochromatic coherent X-ray beam with a high flux of photons. A 34 keV X-ray beam energy was used providing projections with a pixel size of 2.44 µm.

To reproduce real rolling conditions, a high piston velocity is used (1 mm.s -1 ).

It ensures a strain rate of about 1s -1 . Thus, interrupted in situ tests are A total of 18 samples have been tested with different geometries (11 cylinders, 4 double-collars, 2 Brazilians and 1 tensile test). For cylinders, different temperature (from 430 to 520°C), compression speeds (from 0.1 to 1 mm.s -1 ) and alloys have been tested (2XXX and 7XXX series).

Data processing and tracking method

Three phases are distinguishable in the tomographic data: pores, matrix and intermetallics (as visible in Figure 5). Image processing involving filters and thresholding is used to isolate the pores (with ImageJ software [START_REF] Schneider | NIH Image to ImageJ: 25 years of image analysis[END_REF]). Two consecutive backgrounds were subtracted from the image.

The first one is generated using a 2D mean filter with a radius of 20 voxels.

It allows pores to be detected and to fill them to generate a second 2D mean background with a radius of 10 voxels. Then pore thresholding is performed using a cutoff value kept constant for all the samples. In the following, only pores with a volume higher than 8 voxels (or 116 µm 3 ) will be considered. All in all, 18 samples have been scanned with a total of 158 scans. The data processing workflow has been automated to get reliable and comparable data from one scan to another (Figure 6). Image processing and DVC give access to the morphological evolution of every pore in every sample during the compression or tensile tests. Coupling with simulation, which provides local loading path, it is possible to access individual pore closure or opening kinetics, meaning the relationship between morphological evolution and local loading

conditions. An example of such evolution is given in Figure 6 with the pore volume change as a function of cumulated strain. At mid-width (center and quarter thickness) loading state is more compressive and strain increases towards the rolls (quarter thickness). Regardless to the position, the triaxiality as a function of the cumulated strain is almost constant.

Loading paths can be approximated as the final loading state using two descriptors: the final cumulated strain εf (i.e. cumulated strain reached locally at the end of the pass) and the average triaxiality T X f (Equation 7).

T X f = 1 εf εf 0 T X dε (7)
Each considered position in the plate has an influence on both εf and T X f .

For a given position, an increase of relative reduction r results in an increase of When looking at the Lode parameter, two zones are distinguished. Away from the edge, the Lode parameter is zero. This value results from the plane stain state. Thus, in most of the plate variations of pore closure kinetics are not influenced by the Lode parameter which is constant. Close to the edges, the plane strain state is not fulfilled and the Lode parameter takes negative values.

Its absolute value decreases when the cumulated strain increases. Besides, these zones experience low triaxiality (in absolute value). In that case, either in tension or in compression, [START_REF] Zhang | Numerical analysis of the influence of the Lode parameter on void growth[END_REF] and [START_REF] Wang | A void evolution model accounting for stress triaxiality, Lode parameter and effective strain for hot metal forming[END_REF] observed that the Lode parameter has an influence on the pore volume evolution. They also observed that the Lode parameter affects the evolution of the pore shape when observing spherical or ellipsoidal pores. However, in our present case, we want to study real casting pores which have complex shapes. Most of them are tortuous and branched. Contrary to ellipsoid, the shape of a real pore cannot be described with a few number of parameters for which we can determine the sensibility to the Lode parameter. Thus, the evolution of volume due to the modification of the pore shape will depend on the Lode parameter value but also on the initial shape of the pore.

Considering both stress triaxiality and Lode parameter to describe the stress states encountered during rolling, we can identify three different zones:

The edge of the plate (edge and extreme edge positions) and the center of the plate for low deformation where T X ≥ 0 and µ ≤ 0.

The center of the plate for high deformation and the quarter thickness where 0 > T X ≥ -0.7 and µ = 0.

The surface where T X < -0.7 and µ = 0.

Loading paths during in situ tests

Simulations are performed to determine the loading states accessible with uniaxial mechanical tests on small samples. The objective is to find sample geometries that cover the entire map of loading states previously defined (Figure 8.d).

The loading states of uniaxial tests can be compared to those of rolling using the mean triaxiality and the cumulated strain at the end of deformation. However, it is not relevant to do so since the final cumulated strain only depends on the total displacement of the dies imposed during the test. For example, in compression (Figure 10.a), at the center of the cylinder, the final strain reaches a higher value than for a single rolling pass (2 versus 1.2). Nevertheless, all intermediate cumulated strain values up to the final value give access to many mean triaxiality ratios because triaxiality is not constant during deformation for uniaxial tests. Thus, for each sample, the accessible loading states are represented by the envelope of all the loading paths T X = f (ε) of all the elements of the sample. Note that loading paths described as T X = f (ε) are slightly different from those described by T X = f (ε). Figure 10 shows this envelope for the different sample geometries. Results for the simulations for double-collar samples are given in Figure 10.c.

For double-collar geometries, the upper part of the sample acts as an indent for lower part. This results in positive triaxiality ratios even at early stage of deformation.

Finally, with the tensile test presented on Figure 10.d positive triaxiality ratios of about 0.33 can be reached with a low dispersion on loading paths.

With this geometry, there is no need to consider elements at the border of the specimen to access positive stress triaxiality ratios.

23 At the center of the plate at high deformation and the quarter thickness 0 > T X ≥ -0.7 and µ = 0. The range of triaxiality is reproduced by the different types of compression tests. There is a variability of the Lode parameter between 1 and 0 in those samples. [START_REF] Chbihi | Influence of Lode angle on modelling of void closure in hot metal forming processes[END_REF] found that for a given pore at T X = -0.33, the closure strain is 0.6 at µ = 1 and it is 0.75 for µ = 0. On the other hand, Saby et al. (2015a) observed that the closure strain varies by twice on ellipsoids when different orientations or shape factors are considered (for T X = -0.33). In the present case of real casting pores with various orientations and shapes, the effect of the Lode parameter in this range of triaxiality will be of second order compared to the effect of morphology. For this zone we will only focus on the reproduction of triaxiality.

2 1 1 2 2 1 1 2 3 a) b) c) d)
At the surface T X < -0.7 and µ = 0. The triaxiality is highly negative, the closure is controlled by the value of the triaxiality and the Lode parameter has a negligible effect. Such loading conditions are fulfilled at the center of the sample under uniaxial compression.

The combination of the different geometries of uniaxial tests enables to reach all the final loading states of rolling. Note that this study gives all the accessible loading states in in situ samples, however, experimentally pores are not present in every element of every sample. Thus, it is necessary to perform several tests with the same sample geometry to increase the chance to have pores in regions of interest. We can observe several tens of pores in the sample. Figure 12.c shows the loading paths of two clusters of pores together with their morphological evolution.

Example of experimental results

Loading path is obtained with simulation and morphological evolution thanks to DVC. Cluster A which experiences a triaxiality close to zero has few changes in shape whereas cluster B that is subjected to higher negative triaxiality and higher cumulated strain progressively closes.

Such results are accessible for all the pores in all the tested samples. Thus, closure kinetics of real pores with large ranges of triaxiality and cumulated strain, comparable to those of hot rolling, can be studied.

On the choice of loading paths descriptors

The hydrostatic integration (noted Q in Equation 8) is often used to describe pore closure, as reviewed by (Saby et al., 2015b). It is equal to the product of the two loading state descriptors chosen before, namely the average triaxiality and the cumulated strain. The hydrostatic integration takes into account the variation of triaxiality during the test and it is computed locally to account for the inhomogeneities of mechanical fields in the sample. The good qualitative correlation between the hydrostatic integration and the pore shrinkage or growth shows that mean triaxiality and cumulated strain are relevant descriptors to focus on in order to reproduce loading paths.

Q(ε) = ε 0 T X (ε)dε = T X ε (8)

Conclusions

In situ mechanical tests characterized by X-ray microtomography can be used to acquire experimental data on real pore volume evolution under deformation representative of hot rolling. Nevertheless, this method requires small samples (with dimensions of a few millimeters) whereas industrial rolling involves plates at the meter scale. To ensure that the loading states accessible with uniaxial tests are representative of the loading states of rolling, FE simulations of both industrial rolling at the meter scale and uniaxial tests at the millimeter scale were performed. The loading states that can be experienced by pores in terms of triaxiality and cumulated strain were compared. Based on these simulations, the following conclusions can be drawn:

When a single pass of rolling is considered, the use of the mean triaxiality The average triaxiality T X and cumulated strain ε have been used to compare loading paths encountered during rolling and during uniaxial tests. This choice is justified by a qualitative comparison of the hydrostatic integration field, combining both parameters, and the locations where pores shrink or grow.

From the volume evolution of pores with the hydrostatic integration, it can be deduced that pores closure is enhanced by high cumulated strain and high negative triaxiality. They can be reached during rolling with a high relative reduction and high values of L/H ratio.

Thus, with the method described in this article, experimental data on the evolution of real pores are acquired during loading representative of hot rolling.

For the sake of clarity, the conclusions on pore closure are limited in this paper.

Nevertheless, this method can have many applications. It could be used, on the one hand, to obtain a better description of pore closure or opening kinetics by studying pore volume evolution as a function of the hydrostatic integration.

It also gives access to fine morphological evolution in 3D during deformation which will allow mechanisms involved in pore closure or opening to be studied.

Finally, the database of hundreds of pores could be used to test the different models of pore volume evolution available in the literature.

Figure 1 :

 1 Figure 1: Schematic description of a rolling pass. h 1 : entry thickness, h 2 : exit thickness, R cyl : radius of the rolls, L: contact length between the rolls and the plate.

FE

  simulations of uniaxial tests are performed using the software LS-DYNA (LS-DYNA, 2017). Different sample geometries are tested either in compression (cylinder, Brazilian and double-collar) or in tension to broaden the range of accessible loading states (Figure 2). Double-collar geometry was inspired from the work of Farrugia (2006). For cylinders, double-collars and tension samples, 2D axisymmetric elements are used. For Brazilian tests, 3D hexahedral fully integrated elements are used. The material constitutive equations follow the Sellars-Tegart law (Equation

Figure 2 :

 2 Figure 2: Simulated geometries to investigate loading paths under uniaxial loading. Dimensions are given in millimeters. Loading direction is vertical.

  Figure 3.b) to perform in situ mechanical tests at high temperature. The device can perform compressionand tensile tests at a controlled speed (up to 1 mm.s -1 ). The applied load is recorded. The sample is placed between the movable piston and the fixed bell (made of boron nitride to be transparent to X-rays) (Figure3.c). The micropress is mounted on the rotating stage and a furnace is moved down around the bell to heat the sample (Figure3.a). The furnace is independent from the rotation stage and does not move during the tomography acquisition. Two glassy carbon windows are located on the X-ray path at the entrance and exit of the furnace.

Figure 3 :

 3 Figure 3: Experimental setup: a) picture of the micropress and the furnace on the ESRF-ID19 beamline. b) Sketch of the micropress. c) Sketch of the upper part of the micropress composed of the bell and the piston surrounding the sample.

Figure 4 :

 4 Figure 4: Displacement of the die as a function of time for an interrupted compression in 8 steps of 150 µm at 1 mm.s -1 .

Figure 5 :

 5 Figure 5: Visible phases in a cross section of a cylindrical sample after reconstruction: pores (dark), intermetallics (bright) and the matrix (medium grey).

Figure 6 :

 6 Figure 6: Processing workflow to obtain the pore closure kinetics from radiographs acquired with tomography.

Figure 7 Figure 7 :

 77 Figure 7 presents the distribution of hydrostatic pressure within the section located right between the rolls. The thickness is the dimension of the plate along Short-Transverse (ST) direction. The width is along Long-Transverse (LT) direction. High hydrostatic pressure is reached close to the rolls at the midwidth while low hydrostatic pressure is reached on the edges at mid-thickness.In this area of negative hydrostatic pressure, pores will tend to open. Thus, loading path is highly dependent on the considered position in the transverse section.Mechanical fields at five representative positions in the plate are studied (positions marked on Figure7). The center position is located at mid-width and mid-thickness. Two other positions are studied along mid-width: the surface position which is in contact with the rolls and the quarter thickness position which is initially (i.e. before rolling) at mid-way between center and surface positions. Along mid-thickness, extreme edge position is at the side surface of the plate and edge position is where the hydrostatic pressure reaches lowest values during rolling.

Figure 8

 8 Figure 8: a-c) Effect of different parameters on the loading paths of a single rolling pass: a) position in the transverse section, b) relative reduction value, c) L/H ratio value. d) Loading states of rolling to be reproduced.

Figure 8 .Figure 9

 89 Figure 8.d shows the loading states at different positions for passes of dif-

Figure 10 :

 10 Figure 10: Loading domains and local triaxiality for the uniaxial tests: a) compression of a cylinder, b) compression of a double-collar, c) Brazilian compression of a cylinder, d) tension of a notched sample.

Figure 10

 10 Figure 10.b presents the loading domain for Brazilian tests. Large strains at triaxiality close to zero are accessible in the elements at the center of the cylinder base (in element 2). Tension due to the barrel effect is accentuated during the test. High triaxiality ratios are reached, like in element 3, but strain remains low.

Figure 11 :

 11 Figure 11: Loading domains and local Lode parameter for the uniaxial tests: a) compression of a cylinder, b) compression of a double-collar, c) Brazilian compression of a cylinder, d) tension of a notched sample. The local paths are those of elements presented on Figure 10.

Figure 11

 11 Figure 11 presents the evolution of the Lode parameter during uniaxial tests. For deformation in compression (Figure 11.a-c) the value of the Lode parameter is mostly between µ = 0 (representative of plane strain) and µ = 1 (representative of uniaxial stress). With the compression of a cylinder or a double-collar, it is possible to reach negative values for the Lode parameter, in the elements close to the border of the sample. For the tensile test the Lode parameter is close to -1 which is representative of uniaxial tension.

Figure 12

 12 Figure12shows the experimental results for a double-collar sample, with the 3D evolution of pores within the sample (Figure12.a). The relative reduction (r) is computed with the initial and current height of the sample (it is not linked to that of rolling). The strain is inhomogeneous in the sample, the upper part of the sample with a smaller radius deforms more and deformation is progressively

Figure 12 :

 12 Figure 12: Experimental results of the volume evolution of pores in a double-collar sample. a) 3D rendering of the pores (in blue) in the sample during compression. b) Initial state: schematic representation of all the pores (pores are represented by black points with a diameter equal to the pore equivalent diameter, they are projected in the axisymmetric geometry used for simulation) and 3D rendering of two clusters of pores. c) Loading path and morphological evolution of the two considered clusters.

Figure 13 :

 13 Figure 13: Superposition of the hydrostatic integration field Q and the pores present in a cylindrical sample during its compression. Pores are represented by black points with a diameter equal to the pore equivalent diameter. Everything is represented in the initial state geometry and in the axisymmetric plane, taking into account the axial and planar symmetries of the cylinder. In white, tracking of two pores in zone of low |Q|.

Figure 13

 13 Figure13shows the qualitative correlation between the hydrostatic integration and the evolution of the volume of real pores. The colored background rep-

  during deformation and the final cumulated strain gives a good approximation of the local loading state experienced during the pass. Their values depend on the considered position in the plate. The relative reduction of the pass has a direct influence on the cumulated strain and the triaxiality value is linked to the L/H ratio. The combination of several sample geometries deformed uniaxially allows all the loading states experienced during thick plate hot rolling to be reproduced. Selected tests are: compression of a cylinder along its axis, Brazilian test, double-collar compression and tensile test on a notched sample. In cases where the Lode parameter affects pore volume evolution, for low absolute values of triaxiality, those geometries also reproduce the ranges of values of the Lode parameters encountered during rolling.A dedicated sample environment integrated within a synchrotron beamline was developed to capture samples evolution during interrupted in situ tests at high temperature with X-ray microtomography. An automated data processing workflow combining images processing, DVC and FE simulation was set up. It allows the morphological evolution of each pore to be linked to the local mechanical fields it experiences during the deformation.

Acknowledgments

The authors would like to thank the Association Nationale Recherche Technologie (ANRT) and Constellium Technology Center (C-TEC) for funding the project. The European Synchrotron Radiation Facility (ESRF) is gratefully acknowledged for offering beam-time associated to the MA3482 proposal. Thanks to Gary Admans for proofreading the article.

Zhu Rh, Liu Q, Li Jf, Chen Yl, Zhang Xh, Zheng Zq. Flow curve correction and processing map of 2050 Al-Li alloy. Transactions of Nonferrous Metals Society of China 2018;28(3):404-14.


P Gravier

F Mas

A Barthelemy

E Boller

L Salvo

P Lhuissier
email: pierre.lhuissier@simap.grenoble-inp.fr

Mechanisms and kinetics of pore closure in thick aluminum plate

Keywords: Aluminum, Porosity, Closure kinetics, In situ X-ray microtomography, Pore closure model

Interrupted in situ tests at high temperature characterized with X-ray microtomography were employed to study porosity evolution under loading conditions representative of aluminum hot rolling. Coupling digital volume correlation and simulations, it is possible to access, for each pore, its morphological evolution, and the mechanical fields it experienced. The closure kinetics of hundreds of pores were studied, which is described as the pore volume as a function of the hydrostatic integration. Closure kinetics are a function of the shape of the pore.

Focus is put on pores with a complex shape which are more detrimental to mechanical properties. Their closure kinetics are exponential due to the pore fragmentation during closure. The closure of complex pores is fostered by an initial orientation perpendicular to the loading direction and a low initial fragmentation. Based on the closure kinetics of 243 complex pores, a new pore closure model is proposed.

Introduction

Thick aluminum plate is widely used in aerospace applications to machine structural aircraft elements. During casting, porosity appears in the ingot due to the higher density of solid aluminum and the lower hydrogen solubility in solid aluminum compared to liquid aluminum. Pores must be closed during the transformation because their presence in the final product would significantly decrease the mechanical properties of the material. The fatigue life of the material is reduced [START_REF] Couper | Casting Defects and the Fatigue Behaviour of an Aluminium Casting Alloy[END_REF] as well as the ductility and tensile strength [START_REF] Wu | Linking Porosity to Rolling Reduction and Fatigue Lifetime of Hot Rolled AA7xxx Alloys by 3D X-Ray Computed Tomography[END_REF]. Hot rolling is the processing step which enables to close the pores in these thick plates. [START_REF] Park | A study of void crushing in large forgings I: Bonding mechanism and estimation model for bonding efficiency[END_REF] demonstrated that pore closure occurs in two steps.

First morphology and topology of the pore evolve, along with the pore volume decrease. Then, the internal surfaces of the pore weld together during bounding. In the following, we will only focus on the first step also called mechanical closure by Saby et al. (2015b). The present work aims at a better understanding of pore closure mechanisms and at providing a better description of pore volume evolution during pore closure.

From the literature, we can report that mechanical fields expressed through the triaxiality and cumulated strain play a major role in pore closure, they were used by [START_REF] Gurson | Continuum Theory of Ductile Rupture by Void Nucleation and Growth: Part I-Yield Criteria and Flow Rules for Porous Ductile Media[END_REF], [START_REF] Duva | Constitutive potentials for dilutely nonlinear materials[END_REF], [START_REF] Zhang | A criterion for void closure in large ingots during hot forging[END_REF] and Saby et al. (2015a). Besides, closure also depends on pore shape and orientation, as shown by Saby et al. (2015a) who developed a closure model for ellipsoidal pores taking into account the orientation and the shape of the pore. [START_REF] Llanos | Improvement of central soundness in long products from a through process control of solidification and reheating and rolling[END_REF] observed that natural holes are harder to close than artificial drilled holes. Nevertheless, most of the experiments and numerical simulations are conducted on simple pore geometries (spherical or ellipsoidal) whereas real casting pores can be branched and tortuous. The present paper investigates, using in situ X-ray microtomography, the closure of real pores experiencing loading paths representative of hot rolling.

Experimental methods

The experimental methods are detailed in Gravier et al. (under revision -PROTEC-D-21-02245). In summary, in situ mechanical tests at high temperature characterized by X-ray microtomography are performed to mimic hot rolling loading and observe porosity evolution during deformation. Tests are performed in a synchrotron (ESRF-ID19, Grenoble, France), providing volumes with a voxel size of 2.44 µm where the whole sample and pores are visible.

This method provides 3D images of samples. Post treatment based on image processing, Digital Volume Correlation (DVC) and simulations allows the morphological evolution of tens of pores in each sample to be obtained as well as the local loading paths they experienced.

For the following study of pore closure, we will use data provided by the compression of cylindrical samples. Their dimensions are about 2 mm in diameter and 2 mm in height, they are made of 2XXX series alloy in which porosity was purposely increased. The samples are extracted in casting ingot after homogenization. The microstructure is typically made of equiaxed grains with an average grain size of 180 µm and a random texture. In these samples, the volume fraction of porosity ranges between 2.5 × 10 -4 and 8.5 × 10 -4 . The volume fraction of intermetallics is around 3 × 10 -3 . For the compression, loading is applied along the height of the samples (which will be the vertical direction on the following figures). Five samples are compressed at 480 °C with a compression speed of 1000 µm.s -1 , we will refer to these conditions as standard conditions.

Two different temperatures (455 and 520°C) and two lower compression speeds (100 and 500 µm.s -1 ) are also tested.

Porosity description

Each sample typically contains tens of pores (Figure 1.a). However, because of the resolution of tomography, the precise number of pores is difficult to determine unambiguously, as apparently separate pores may in reality be connected by unresolved thin ramifications. Also, some of the pores are close to each other and may interact during deformation (Figure 1.b). In order to describe precisely the tomographic observations, we will use here the concepts of clusters and subpores. Subpores are all the pores that are separate from one another after thresholding (Figure 1.c). When these subpores are less than 50 µm apart (or 20 voxel surface to surface distance), they belong to the same cluster (Figure 1.d). The distance of 50 µm was chosen such that big clusters do not merge. It should be noted that the standardized dye penetrant inspection, used to validate industrial plate production, quantifies objects similar to these clusters: [START_REF] Kishimoto | A numerical investigation of the interaction and coalescence of twin coplanar semi-elliptical fatigue cracks[END_REF] demonstrated that if pores are close to each other, they can be considered as the same pore. The porosity is composed by the sum of all the clusters contained inside the sample. If we are able to predict the volume evolution of each single cluster, we are then able to compute the volume evolution of porosity.

A cluster can be composed of one or more subpores. A cluster's fragmentation is defined as the number of subpores it contains. As the tomographic resolution might be too coarse to image thin ramifications between subpores, fragmentation tends to be overestimated. Nevertheless, subpores that appear separate due to the resolution limitations but are actually connected will be contained within the same cluster. Thus, in this paper the volume evolution of pores will be investigated at the scale of clusters. The pores present in the samples have various shapes. They appear at solidification due to two concurrent mechanisms as reported by [START_REF] Dantzig | Solidification: Revised and Expanded[END_REF]. On the one hand, the lack of mass feeding to compensate for the volumetric shrinkage causes the formation of pores in the interdendritic region. These pores are branched and tortuous. On the other hand, the lower hydrogen solubility in solid aluminum compared to liquid aluminum results in the liquid enrichment in gas. The gas is rejected under the shape of bubbles forming spherical pores. Shape has been reported to have a strong influence on pore closure. [START_REF] Llanos | Improvement of central soundness in long products from a through process control of solidification and reheating and rolling[END_REF] observed differences between artificial and drilled pores, [START_REF] Kb | Simulation of void defect evolvement during the forging of steel ingot[END_REF] reported differences in the simulated closure of tetrahedral and spherical pores and Saby et al. (2015a) took into account shape factors in the closure of ellipsoids. Based on these observations, pores are classified into 5 categories: spherical, equiaxed, rod, sheet and complex. The shape classification is adapted from the work of [START_REF] Fang | Autonomous filling of creep cavities in Fe-Au alloys studied by synchrotron X-ray nano-tomography[END_REF] and is applied to subpores. Shapes are determined as a function of three parameters (Tab. 1):

pore elongation E (Eq. 1), flatness F (Eq. 2) and complexity C (Eq. 3).

E = 2a b + c
(1)

F = b c (2)
Where a > b > c are the semi-axes of the pore equivalent ellipsoid, namely the ellipsoid with the same inertia matrix (see Figure 2).

a) b)

Figure 2: a) 3D rendering of a pore (in blue) with its equivalent ellipsoid (in red) that have the same inertia matrix as the pore. b) Semi-axes of the equivalent ellipsoid, a > b > c.

C = 1 s = 1 6V A 3 π (3)
s is the sphericity defined as the ratio of the pore volume V over the volume of a sphere having the same surface area A. For a sphere s = 1, s < 1 otherwise.

Shape

E F C Color code Spherical / / C ≤ 1.5 blue Equiaxed < 4 < 5 1.5 < C < 2.5 red Rod ≥ 4 < 5 1.5 < C < 2.5 green Sheet / ≥ 5 1.5 < C < 2.5 yellow Complex / / C ≥ 2.

pink

Table 1: Shape classification spherical equiaxed rod complex Examples of subpores of each category are shown in Figure 3, no flat subpores corresponding to sheet category were found in the present studied samples.

Color code defined in table 1 will be used in all figures when shape classification is involved. In the following, simple shapes will refer to shapes with a low complexity, namely spherical, equiaxed and rod shapes.

A shape is also attributed to each cluster. Given that a cluster can be composed of subpores with different shapes, the cluster shape is the shape which is the most represented in volume fraction.

The size of a subpore (or a cluster) will be described with the equivalent diameter D eq which is the diameter of a sphere having the same volume V as the subpore (or cluster) (Equation 4).

D eq = 2 3 4π V 1 3 (4)
Finally, orientation is defined at the cluster scale using equivalent ellipsoid.

The semi-axes of the ellipsoid (r 1 > r 2 > r 3 ) and their respective direction

( -→ u 1 , -→ u 2 , -→ u 3
) are computed from the inertia matrix of the cluster. Cluster orientation is the value of the angle between the direction of the major axis ( -→ u 1 ) and the direction of macroscopic loading (vertical direction in the following figures).

Results and discussion

4.1. Shape and size of subpores in the initial state Conversely, spherical subpores have a low equivalent diameter, generally under 20 µm (Figure 4.a). Note that no quantification of subpores was performed below a volume of 8 voxels to avoid artifacts due to image noise. At such a small volume (equivalent diameter of 6 µm), the distribution of voxels cannot form a complex shape, so small pores are more likely to appear spherical due to resolution limits. Equiaxed and rod subpores have intermediate sizes between those of spherical and complex subpores.

Even if spherical subpores are more numerous, complex subpores represent 88% of the total volume of porosity (Figure 4.c). Complex subpores represent most of the porosity. Furthermore, they are the biggest, making them potentially more harmful for mechanical properties. [START_REF] Couper | Casting Defects and the Fatigue Behaviour of an Aluminium Casting Alloy[END_REF] demonstrated that the fatigue life of the material is increased when pores are smaller.

Based on these observations, the description of closure of complex clusters is a key issue.

Shape evolution of subpores during closure

During the compression of cylindrical samples, porosity decreases because of negative triaxiality in most of the volume and large plastic deformation. Due to friction with the dies, the deformation is not homogeneous in the sample: zones close to the dies hardly deform and the sample takes a barrel shape. It induces tensile stresses on the sides of the samples which prevent pore closure and even lead to pore growth or pore nucleation. Therefore, at the end of the compression, pores are closed at the center of the sample and some pores may remain on the edges or may even appear in the last step of compression. At every compression step, a shape was assigned to each subpore. This shape is independent of the subpore history meaning that a subpore can have Position is represented in the initial state using the estimated initial position given by DVC.

The deformation of the sample is not represented to ease the visualization of tracking.

The number of subpores can increase due to fragmentation or nucleation, or 155 decrease due to total closure or coalescence. The number of complex subpores remains constant during the early stage of deformation but their average equivalent diameter decreases sharply. This can be explained by the fragmentation of complex pores into a smaller complex pore and other simpler shapes such as equiaxed pores. The increase in the number of equiaxed pores supports this argument. On the other hand, the number of spherical pores decreases at the beginning of compression meaning that there is more spherical pores that close than spherical pores that appear due to fragmentation. After a strain of 0.3, the number of equiaxed pores reaches a plateau: there is an equilibrium between the creation of equiaxed pores from the fragmentation of bigger complex pores and the fragmentation of equiaxed pores into spherical pores. This fragmentation of large complex subpores to smaller subpores of simpler shapes is visible on Figure 6.c. Subpores are projected on the axisymmetric plane. For each deformation step, subpore position is represented in the initial state using the estimated initial position given by DVC. Thus, deformation of the sample is not represented. Using initial configuration representation helps subpore tracking: if subpores are located at the same position at different deformation steps, they are likely to be the same subpore. Subpores are represented by disks of equivalent diameter (meaning that the disk diameter is equal to the subpore equivalent diameter). Disk color describes the associated subpore shape. Complex subpores decrease in size and split into new equiaxed, rod or spherical subpores. Equiaxed subpores become spherical and finally close. This Figure also shows the nucleation of small spherical subpores close to the edges.

It is correlated with the final increase in number of spherical subpores. Finally, subpores located near the dies do not close because of low strain in this zone.

Morphology evolution of clusters during closure

The evolution of individual cluster morphology has been characterized by the cluster fragmentation and the shape evolution of the subpores in the cluster.

For spherical clusters, there is no fragmentation. Cluster are generally com-posed of a single subpore. The subpore closes progressively, remaining spherical or sometimes becoming equiaxed due to flattening. The final description of spherical pore closure is limited by the resolution of tomography. In contrast, complex clusters are initially composed of several subpores. They are generally made of a central big complex subpore surrounded by smaller subpores. It is possible that these subpores actually connect but that the resolution limit does not enable to see the connection. An example is given on Figure 7. In the present case, the cluster is initially composed of 2 complex subpores with the biggest at the center. They are surrounded by 8 smaller subpores (rod, equiaxed or spherical). The figure describes the evolution of the composition of the cluster during the test. After the first compression step, the smallest complex subpore (at the upper left) fragments into a rod and a spherical subpores. The main complex subpore fragments also but generates only small subpores with simple shapes (from scan 1 to 3). Most of the volume remains in the complex subpore until scan 4. From then on, the big complex subpore fragments into 4 smaller complex subpores of similar volumes and other simple subpores. From scan 6, these complex subpores evolve to simpler shapes and finally close. The overall number of subpores first increases, owing to fragmentation and then decreases due to pore closure.

3D renderings also show that when the cluster has lost most of its volume, its dimension is still large. At scan 4, only 24% of the pore volume remains whereas its dimension perpendicular to the compression direction seems to remain unchanged. The cluster can still be detrimental for mechanical properties even when it reaches a low relative volume because, as observed by [START_REF] Murakami | Effects of defects, inclusions and inhomogeneities on fatigue strength[END_REF], the pore has a crack shape. Therefore, it is necessary to describe finely the final closure of clusters up to a very small remaining volume.

Closure kinetics

According to literature, triaxiality T X and cumulated strain ε are the two main parameters controlling pore closure. The hydrostatic integration Q (Equa-tion 5) proposed by [START_REF] Tanaka | Factors contributing to crushing of voids during forging[END_REF] combines these parameters. Qualitatively, pore volume evolution correlates well with Q. Pore closure kinetics will be described by the evolution of the volume of pore clusters as a function of the hydrostatic integration Q. A large scatter in closure kinetics is observed. The influence of morphological parameters is investigated to determine whether they can explain this scatter.

Q(ε) = ε 0 T X (ε)dε = T X ε (
First, shape has an influence on pore closure kinetics. Figure 8.b presents the closure kinetic of all complex clusters in a given sample. Figure 8.c presents the closure kinetics of spherical clusters in the same sample. As spherical clusters are smaller than complex clusters, the relative uncertainty in their volume measurement is higher. Only their initial closure can be described as they reach the minimal volume of 8 voxels for a higher normalized volume. For complex clusters, the volume decreases exponentially with Q whereas spherical clusters closure is more linear or quadratic with Q. This difference could be explained by the fact that complex subpores fragment whereas spherical subpores remain a unique entity during closure. [START_REF] Saby | Compréhension et modélisation des mécanismes de refermeture de porosité dans les procédés de mise en forme des métaux à chaud[END_REF] reported that fragmentation slows down pore closure. In our case, fragmentation happens even in the early stage of closure (Figure 7) when closure rate is still high and it rather seems that the kinetics slow down when the biggest subpore of the cluster fragments in several subpores of similar sizes.

An accurate description of the volume evolution of complex clusters is more important than for spherical clusters since complex clusters are the largest and the most detrimental for mechanical properties. We propose to describe complex cluster closure kinetics using an exponential relation with the hydrostatic integration. For each complex cluster, the following formula was fitted, determining an exponential coefficient A per cluster.

V V 0 = exp (AQ) (6)
Q being negative in compression and for positive values of A, this curve decreases with cumulated strain. Figure 8.d shows the fitting curve for two complex clusters. The higher the value of A, the faster the closure kinetics of the pore.

The proposed fit can be expressed by the following derivative form:

V V = A T X ε (7)
It is the same expression as the analytical volume evolution of a spherical pore in a linear viscous matrix calculated by [START_REF] Budiansky | Void Growth and Collapse in Viscous Solids[END_REF] where A = 9/4.

The fit proposed here tends to slightly overestimate initial closure and underestimate final closure but enables cluster closure ability to be easily sorted.

The exponential coefficient is determined for all complex pores in the 9 cylinder samples. If the coefficient of determination r 2 is lower than 0.95, the pore is not taken into account. Such low coefficient of determination occurs for pores that are located in the zone of positive triaxiality or close to it. When triaxiality becomes positive, the hydrostatic integration which was decreasing starts to increase whereas pore volume continues to decrease before increasing. Misfits also result from low negative triaxiality. In that case, cluster volume keeps decreasing whereas hydrostatic integration decreases slower due to low nega- The following observations can be made:

304

A correlation between A and the equivalent diameter D eq is not obvious 305 20 and thus, in the range of volume tested, pore closure does not depend on pore size. This observation is consistent with results of [START_REF] Saby | Compréhension et modélisation des mécanismes de refermeture de porosité dans les procédés de mise en forme des métaux à chaud[END_REF] and [START_REF] Chen | Numerical simulation and experimental verification of void evolution inside large forgings during hot working[END_REF].

Higher values of A are reached for clusters with a low initial fragmentation.

In this case, interaction between individual subpores in a cluster would not be beneficial for closure. This is consistent with the results of Saby ( 2013) who observed that fragmentation slows down the closure kinetics.

A reaches the highest values for clusters oriented perpendicularly to the deformation direction. This result is consistent with those of [START_REF] Saby | Compréhension et modélisation des mécanismes de refermeture de porosité dans les procédés de mise en forme des métaux à chaud[END_REF] where faster closure was observed for ellipsoids compressed along their smallest semi-axis (so compression direction forms an angle of 90°with the longest semi-axis). It is also consistent with the model of [START_REF] Feng | A 3-D model for void evolution in viscous materials under large compressive deformation[END_REF] which predicts faster closure for ellipsoids that have a low dimension in the compression direction. The orientation effect is accentuated by large aspect ratios (r 1 /r 3 ). Here, only orientation is taken into account and not the ellipsoid aspect ratio. Thus, pores with low aspect ratio can be oriented perpendicularly to the compression and result in a low A value.

Note that the notions of orientation and aspect ratio are more difficult to define for a real tortuous pore than for an ellipsoid.

Conclusions

In this work we were able to study porosity closure based on the observation of pores during their deformation using X-ray microtomography. These interrupted in situ tests were performed under conditions representative of hot rolling and the closure of several hundreds of real pores was followed. Pores were studied at two scales (subpores and clusters) and as a function of their shape (spherical, rod, equiaxed and complex). Based on this work, the following conclusions can be drawn:

Complex clusters are bigger in size and represent the largest part of the porosity compared to pores with other shapes. The accuracy of the overall porosity prediction is mainly linked to the ability to properly describe complex subpores closure.

During closure, complex subpores fragment into smaller subpores with simpler shapes.

Closure kinetics strongly depend on cluster shape. Closure kinetics of complex clusters can be approximated by an exponential function of the hydrostatic integration whereas a linear or quadratic dependency on the hydrostatic integration is more suitable in the case of spherical clusters.

For complex clusters, closure is influenced by the principal orientation and the initial fragmentation of the cluster. Clusters oriented perpendicularly to the compression direction and with low initial fragmentation are easier to close.

Based on experimental data analysis, a new closure model is proposed:

V /V 0 = exp(9.1 Q). It only depends on the hydrostatic integration. It aims at properly describing complex pore closure up to relative volume far below 20% because pore dimensions are still large for such a remaining volume.

This study proposes a prediction model for pore closure by fitting experimental data. Nevertheless, many other models are available in the literature.

Some of these models will be tested in a future paper. These models can be computed using the present data collected with X-ray microtomography and with the local mechanical fields provided by simulations.

Figure 1

 1 Figure 1: a) 3D rendering of pores (in blue) within a cylindrical sample. b) Zoom on pores. c) Pores colored by subpores: each individual pore has a different color. d) Pores colored by clusters: pores less than 50µm apart belong to the same cluster and have the same color.

Figure 3 :

 3 Figure 3: Examples of 3D views of different pore shapes according to the classification defined in table 1. Note that according to this classification, spherical pores can have protrusions and equiaxed pores can have small branches.

Figure 4 Figure 4 :

 44 Figure4shows the distribution of subpores as a function of their shape and their equivalent diameter. These are subpores present in the initial state in a given sample.

Figure 5 :

 5 Figure 5: 3D rendering of pores (in blue) within a cylindrical sample during its compression.

  different shapes at different deformation steps. For each shape category, average equivalent diameter and number of subpores are plotted in Figures 6.a. and b.

Figure 6 :

 6 Figure 6: Evolution in size and number of subpores during compression depending on their shape. a) Evolution of the average equivalent diameter per shape. b) Evolution of the number of subpores per shape. c) Subpores position, size and shape in the axisymmetric projection.

  14

Figure 7 :

 7 Figure 7: Fragmentation of a complex cluster during closure. a) 3D rendering of the cluster at different compression steps (loading was applied in the vertical direction). b) Schematic representation of the subpores composing the cluster.

  5)Recent pore closure models developed by[START_REF] Chbihi | Influence of Lode angle on modelling of void closure in hot metal forming processes[END_REF],[START_REF] Zhang | A porosity closure model considering stress triaxiality ratio and Lode stress parameter[END_REF],[START_REF] Wang | A void evolution model accounting for stress triaxiality, Lode parameter and effective strain for hot metal forming[END_REF] take into account the Lode parameter µ in addition to the triaxiality. The influence of the Lode parameter affects the evolution of the pore volume when the traixiality is close to zero. In the present study, T X ≤ -0.1 and 0 ≤ µ ≤ 1.[START_REF] Wang | A void evolution model accounting for stress triaxiality, Lode parameter and effective strain for hot metal forming[END_REF] found that for a given pore at T X = -0.3, the closure strain is 0.55 for µ = 1 and it is 0.75 for µ = 0. On the other hand,Saby et al. (2015a) observed that the closure strain varies by twice on ellipsoids when different orientations or shape factors are considered. In the present case of real casting pores, the effect of the Lode parameter will be neglected compared to the effect of morphology. Thus we will not take the Lode parameter into account to study closure kinetics. The pores studied here experience stress states representative of the center of the plate or close to the rolls during rolling. They are not representative of the edges of the plate where the triaxiality is positive or close to zero. More details on the reproduction of stress states are given in Gravier et al. (under revision -PROTEC-D-21-02245).Closure kinetics are studied observing clusters in compressive zones of cylindrical samples deformed by compression. Closure kinetics of all the clusters in a given cylindrical sample are presented on Figure8.a. As hydrostatic integration is negative and decreases along deformation, -Q is used for the X-axis. Only points with non-zero volume are represented on these graphs. When volume is equal to zero it is not possible to determine if clusters are really closed or if they are no longer detected (lost during tracking or volume lower than detection limit of 8 voxels). In the following, clusters with an initial volume of less than 16 40 voxels (= 581 µm 3 ) were not considered because their volume is lower than 8 voxels when they reach 20% of their initial volume. For such small clusters, final closure cannot be studied.

Figure 8 :

 8 Figure 8: Cluster closure kinetics in a sample. a) All clusters. b) All complex clusters with an example of morphology evolution. c) All spherical clusters with an example of morphology evolution. d) Example of the exponential fit on two complex clusters and 3D rendering of their initial shape.

  tive triaxiality. It results in a too high negative slope compared to those of the exponential formula. These observations show the limitation of hydrostatic integration in the description of loading path. In total 243 exponential coefficients A are fitted on closure kinetics of real complex clusters. Distribution of all the values obtained is presented on Figure 9.a. Different parameters are tested to draw possible correlations between the exponential coefficients of a cluster and morphological and mechanical parameters. Some results are presented on Figures 9.b-d, all coefficients fitted for complex clusters in cylindrical compressed samples are gathered on this figure.

Figure 9 :

 9 Figure 9: Study of the exponential coefficient A. a) Distribution of the values of A fitted on 243 closure kinetics of complex clusters. b-c) Dependency of exponential coefficient to several morphological parameters: equivalent diameter (b), orientation (c) and initial fragmentation (d).

  If the pore is branched, with arms in different directions, global orientation of the pore does not faithfully represent the different arms. It would be more relevant to adopt a more local description of the pore orientation. As a conclusion, complex cluster closure kinetics are mostly influenced by the initial fragmentation of clusters and their orientations. It is difficult to determine the initial fragmentation and orientation of pores in an industrial plate prior to rolling. Thus, we propose a new closure model expressed by Equation 6 with a fixed value of A = 9.1 for all clusters. This phenomenological description represents the average closure behavior of studied complex clusters and captures the closure kinetics induced by fragmentation. 4.5. Effects of temperature and compression speed on closure kinetics The previous results show data acquired during the compression of samples at different temperatures (455, 480 and 520°C) and speeds (100, 500 and 1000 µm.s -1 ). FEM simulations have shown that the local loading paths, described by the evolution of stress triaxiality as a function of the cumulated strain, are not influenced by the temperature or speed Gravier et al. (under revision -PROTEC-D-21-02245). Thus, closure kinetics were considered independently of the compression conditions.

Figure 10

 10 Figure10shows the distribution, under the shape of a boxplot, of the exponential coefficients in each studied sample (ends of the whiskers correspond to the largest points that fall within a distance of less than 1.5 times the interquartile range over the third quartile and under the first quartile). If the experimental conditions used to test a sample act in favor of pore closure, the distribution would be shifted toward higher A values. The figure shows that discrepancies between standard conditions and tests with different temperatures and speeds are less significant than discrepancies between samples tested with same standard conditions. It means that temperature and compression speed have a second order influence on pore closure compared to cluster morphology such as orientation or fragmentation.

Figure 10 :

 10 Figure 10: Boxplots of the distribution of exponential coefficient for the different samples.

Acknowledgments

The authors would like to thank the Association Nationale Recherche Technologie (ANRT) and Constellium Technology Center (C-TEC) for funding the project. The European Synchrotron Radiation Facility (ESRF) is gratefully acknowledged for offering beam-time associated to the MA3482 proposal. Thanks to Gary Admans for proofreading the article.