domain: sde sdu.ocean spi

text:
François Mathé

Note technique CONFORMITÉ TECHNIQUE D'APPAREILLAGES POUR LA SURVEILLANCE RÉGLEMENTAIRE DE LA QUALITÉ DE L'AIR MODALITES D'EVALUATION DES DISPOSITIFS DE MESURE

SYNTHESE

Le présent document détaille le processus d'évaluation de la conformité technique des dispositifs de mesure pour la surveillance réglementaire de la qualité de l'air ambiant vis-à-vis des exigences fixées par le LCSQA, en tant que Laboratoire National de Référence notifié par le ministère en charge de l'environnement. Les rôles et responsabilités des différents acteurs (ministère en charge de l'environnement, LCSQA, AASQA, demandeur de l'évaluation de conformité technique) sont décrits.

Introduction

En France, conformément au Code de l'Environnement (articles L221-1 à 10), la mise en oeuvre de la surveillance de la qualité de l'air et l'information du public sont confiées par l'État à des Associations Agréées de Surveillance de la Qualité de l'Air (AASQA). L'agrément est délivré par le Ministère en charge de l'environnement qui élabore notamment les politiques de surveillance de la qualité de l'air à mettre en oeuvre. Les AASQA, toutes régionalisées depuis le 1 er janvier 2012, sont des associations multipartites régies par la loi du 1 er juillet 1901 relative au contrat d'association.

La coordination technique du dispositif de surveillance de la qualité de l'air est confiée au Laboratoire Central de Surveillance de la Qualité de l'air (LCSQA). Le LCSQA est un Groupement d'Intérêt Scientifique qui regroupe trois partenaires : l'Institut National de l'Environnement Industriel et des Risques (INERIS), le Laboratoire National de métrologie et d'Essais (LNE) et l'Ecole nationale supérieure Mines-Télécom Nord Europe (IMT NE). Le LCSQA est le Laboratoire National de Référence notifié [START_REF] Nf En | Qualité de l'air ambiant -Méthode normalisée pour le mesurage de la concentration en dioxyde d'azote NO2 et monoxyde d'azote NO par chimiluminescence IR[END_REF] par le Ministère en charge de l'environnement, en réponse aux exigences réglementaires européennes. Il a un rôle de référent technique au niveau national et apporte un appui stratégique, technique et scientifique au dispositif. Dans le cadre du processus de vérification de la conformité des appareillages dans le présent document, le LCSQA assure la coordination technique du processus de vérification de la conformité technique.

Contexte réglementaire

Les référentiels réglementaires applicables en France pour la surveillance de la qualité de l'air sont le Code de l'Environnement (Titre II -Livre II) et ses textes d'application et notamment l'arrêté du 16 avril 2021 relatif au dispositif national de surveillance de la qualité de l'air ambiant. Ils assurent la transcription des textes européens :

-la Directive n° 2004/107/CE du 15

Modalités générales du processus de vérification de conformité des appareils

Dans le cadre de leurs activités au sein du Laboratoire National de Référence (tel que prescrit dans l'arrêté du 16/04/21 -Titre II -et notifié par le Ministère en charge de l'environnement) et de son rôle de coordinateur technique de la surveillance de la qualité de l'air, les membres du LCSQA ont proposé au Ministère, qui l'a approuvé, un processus de vérification de conformité technique des appareillages utilisés par les AASQA pour la mesure de polluants atmosphériques réglementés.

Il est prévu que le même processus de vérification de conformité soit étendu à tout appareil de mesure utile au Dispositif National français (ex : système de mesure indicative), à la demande d'un fournisseur ou du Ministère en charge de l'environnement.

Le constat de conformité est à terme destiné à tout dispositif de mesure de polluant de l'air ambiant en référence aux Directives et textes français, c'est à dire tout dispositif (électrique ou non) utilisé dans la chaîne de mesure (du prélèvement au rapatriement de données en Poste Central) pour la surveillance de la qualité de l'air.

Pour répondre aux exigences européennes (notamment l'annexe VI.D « introduction de nouveaux appareils » de la Directive 2008/50/CE), les appareils automatiques, les préleveurs (gaz et/ou particules) et les collecteurs de précipitation sont les dispositifs pris en considération.

Une demande de vérification de conformité peut être faite par un constructeur ou par un distributeur désigné par lui sur le territoire français. Dans le cas du distributeur, les responsabilités liées au processus de vérification de conformité lui incombent (cf. § 2.3).

Le dossier de demande de vérification de conformité technique est étudié par les membres du LCSQA, en collaboration avec la Commission de Suivi « Suivi Technique des AASQA » et, le cas échéant, toute autre Commission de Suivi en fonction des besoins. En fonction des éléments du dossier, un avis technique détaillé sur la demande de vérification de conformité est émis par le LCSQA et transmis au Comité de Pilotage du dispositif de la Surveillance de la qualité de l'air (CPS, auquel participent le Ministère en charge de l'environnement et les représentants des AASQA) pour avis. La reconnaissance de la conformité technique de l'appareillage s'appuie sur cet avis. Un appareillage bénéficie de cette conformité dans la configuration technique de fonctionnement et d'utilisation qui lui a permis de répondre aux exigences techniques du référentiel d'étude (norme, guide…). Cela permet de spécifier le contexte technique d'utilisation du dispositif (ex : station climatisée, moyen mobile…) et d'assigner le statut de mesure (ex : mesure fixe, mesure indicative…). D'autres exigences peuvent également être prises en compte (ex : la gestion de sources radioactives couverte par les exigences de l'Autorité de Sûreté Nucléaire -ASN).

Un suivi technique du constat de conformité est assuré par les membres du LCSQA avec l'aide du demandeur (cf. § 3.5) et des AASQA (ex : au travers de retours d'expérience des utilisateurs, des tests de suivi d'équivalence assuré par le LCSQA…). Ainsi, dans le cas de méthodes dites « équivalentes », les résultats du suivi en continu de l'équivalence géré au niveau national par le LCSQA sont pris en compte dans le processus de vérification de conformité technique pour contrôler et valider l'adéquation des conditions spécifiques de sites rencontrés lors du suivi et celles de la démonstration d'équivalence initiale. Le suivi d'équivalence à l'échelle nationale intègre les typologies de site5 et les conditions climatiques représentatives de la France. Toute modification (technique, informatique) d'un appareil bénéficiant d'un constat de conformité technique implique à minima l'information du LCSQA qui donnera au Ministère en charge de l'environnement un avis technique sur l'impact de la modification sur le constat de conformité technique de l'appareillage (cf. § 3.5), après concertation avec la Commission de Suivi « Suivi Technique des AASQA » et, le cas échéant, toute autre Commission de Suivi en fonction des besoins.

ROLES ET RESPONSABILITES

Le ministère en charge de l'environnement

Le Ministère en charge de l'environnement est responsable de l'application des directives européennes sur le territoire français et, à ce titre, vérifie le respect des exigences associées. Les modalités de cette application sont décrites dans l'arrêté du 16 avril 2021 relatif au dispositif national de surveillance de la qualité de l'air ambiant. Dans le cadre du processus de vérification de conformité technique des appareillages de mesure, le Ministère en charge de l'environnement approuve (ou non) les dispositifs lors du CPS. Si besoin, il se réserve la possibilité de différer le démarrage du processus.

Les membres du LCSQA

Les membres du LCSQA ont en charge l'instruction de la demande de vérification de conformité technique, émettent un avis à destination des participants au processus (sur la base des informations mises à sa disposition) et assurent la gestion de la vérification de conformité technique (de la demande initiale à l'établissement du certificat de conformité ainsi qu'au suivi en continu une fois la conformité technique octroyée) de la manière suivante :

-échanges avec le demandeur sur la complétude du dossier et d'éventuelles demandes complémentaires, -information des participants au processus de vérification de conformité technique, -déclaration du statut de conformité technique via la mention de l'appareil concerné Le demandeur de constat de conformité technique doit s'assurer de la maîtrise de la qualité de la fabrication de l'appareillage concerné, afin que tous les appareillages mis sur le marché français une fois la conformité constatée continuent à respecter les critères de performance appropriés. Il doit également s'assurer de la compatibilité avec la chaîne nationale de transmission de données (dans sa configuration au moment de la demande initiale) sans surcoût pour l'utilisateur. Enfin, il doit indiquer la durée pendant laquelle il est prévisible que les pièces indispensables à l'utilisation de l'appareillage concerné seront disponibles sur le marché, depuis le constat de conformité technique initial jusqu'à l'arrêt de sa commercialisation. Une durée minimale de

(

DETAIL DU PROCESSUS

Dossier initial

Le dossier initial de demande de constat de conformité technique remis par le demandeur (sous format électronique) doit à minima contenir les éléments suivants : Note : Outre les éléments techniques permettant de juger la conformité de l'appareillage par rapport aux exigences demandées, il convient que le rapport donne un descriptif suffisant permettant de juger de la "représentativité" des sites employés lors des tests, afin de pouvoir s'assurer que les conditions de tests sont comparables à celles rencontrées par les AASQA sur le territoire national  le manuel technique de l'appareil avec des procédures QA/QC permettant a minima de répondre aux exigences demandées (en terme de QA/QC en continu pour les AASQA, par exemple, l'étalonnage avec les recommandations du constructeur en terme de moyen et de fréquence). En lien avec la configuration technique, la compatibilité avec tout Système d'Acquisition de Mesures utilisable en AASQA devra être garantie par le demandeur de constat de conformité technique (avec l'appui du constructeur le cas échéant) ;  la liste de pièces détachées usuelles avec les prix associés ;  une estimation de coûts annuels (voire pluri annuels ex: sur 5 ans minimum) de fonctionnement ainsi que des consommables primordiaux pour une utilisation usuelle en AASQA (avec estimation des temps d'intervention). La consommation énergétique sera également à mentionner ;  l'engagement par écrit d'une garantie minimale de 24 mois (pièces et main d'oeuvre, à partir de la date de livraison) ;


 l'engagement par écrit sur un délai minimum et pour une durée spécifiée concernant la fourniture en pièces détachées, l'intervention pour résolution des dysfonctionnements, la reprise d'appareil en fin de vie ;  l'engagement par écrit à une information du LCSQA de toute modification (aussi bien de la partie matérielle -« hard » -que de la partie logicielle -« soft ») sur l'appareil ;

Note : Toute modification implique une demande auprès du LCSQA de réexamen de la conformité technique (dont la durée de traitement dépendra de la nature de la modification). Toute modification susceptible de modifier les performances métrologiques de l'appareil (que ce soit de

Etapes du processus

Tests complémentaires

Les tests complémentaires sont destinés à apporter des éléments d'information additionnels sur les exigences demandées (ex : critères de performance) dans le cas où les membres du LCSQA estiment que les éléments du dossier technique initial ne correspondent pas aux conditions d'exploitation usuelles rencontrées par les AASQA. Certains résultats sont jugés primordiaux tels que la disponibilité de l'appareillage8 ou sa période de fonctionnement sans surveillance [START_REF] Nf | EN 12341 -Air ambiant -Méthode normalisée de mesurage gravimétrique pour la détermination de la concentration massique MP10 ou MP2,5 de matière particulaire en suspension[END_REF] . Les critères de conformité sont les suivants :

 disponibilité de l'appareil :  90% (à calculer selon la formule du § 7.5.6, [17], cf. annexe I) ;  période de fonctionnement sans surveillance : au moins 14 jours.

Chronologie du processus

L'intégralité du processus de vérification de conformité initial ne doit pas excéder 15 mois, sous réserve de la complétude du dossier et des résultats d'essais complémentaires jugés nécessaires par les membres du LCSQA (en collaboration avec la Commission de Suivi « Suivi Technique des AASQA »).

Une fois la conformité constatée, son suivi est assuré sur la base d'un cycle annuel.

Modification d'un appareillage bénéficiant du constat de conformité

Une modification peut être d'ordre technique (ex : changement de pièce) ou informatique (modification de software nécessaire au fonctionnement de l'appareil).

Le demandeur doit pouvoir détailler au moins les informations suivantes sur l'appareillage bénéficiant du constat de conformité :  toute modification datée de composant(s) ;  l'évaluation de l'impact de modification sur les performances demandées ;  les données se rapportant à l'évaluation de l'impact de modification ;  l'évaluation de modification jugée significative (cf. ci-après).

  tout rapport de conformité technique (démonstration de l'équivalence à la méthode de référence, tests d'approbation par type,...) sur la base d'essais effectués par un laboratoire accrédité NF EN ISO/CEI 17025 pour ce type d'activités ou travaillant dans le respect de ses exigences. La configuration technique de l'appareil et de son exploitation devra être clairement détaillée de façon à voir si cette dernière correspond au cas usuel français (par exemple environnement d'exploitation, type d'enregistrement de données…). Ceci peut amener à une limitation dans la configuration d'utilisation ultérieure (par exemple, usage restreint à la station fixe climatisée), voire à des tests complémentaires effectués par (ou en collaboration avec) les membres du LCSQA ;

ANNEXE I : --- 5 -

 :5 Polluants couverts par le schéma de Vérification de Conformité Technique (VCT) d'appareillages (liste au 30/01/2023) Monoxyde d'azote (NO) Dioxyde d'azote (NO2) Oxydes d'azote (NOx) Métaux lourds dans les PM10 :  Plomb (Pb)  Arsenic (As)  Cadmium (Cd)  Nickel (Ni) -Hydrocarbures Aromatiques Polycycliques (HAP) dans les PM10 :  Benzo [a] pyrène (B[a]P)  Benzo [a] anthracène (B [a] A)  Benzo [b] fluoranthène (B [b] F)  Benzo [j] fluoranthène (B [j] F)  Benzo [k] fluoranthène (B [k] F)  Indéno [1,2,3-cd] pyrène (IP)  Dibenz [a, h] anthracène (DB [a, h] A) -Métaux lourds dans les dépôts totaux :  Arsenic (As)  Cadmium (Cd)  Mercure solide ou gazeux (Hg)  Nickel (Ni)  Plomb (Pb) -Hydrocarbures Aromatiques Polycycliques (HAP) dans les dépôts totaux :  Benzo [a] pyrène (B[a]P)  Benzo [a] anthracène (B [a] A)  Benzo [b] fluoranthène (B [b] F)  Benzo [j] fluoranthène (B [j] F)  Benzo [k] fluoranthène (B [k] F)  Indéno [1,2,3-cd] pyrène (IP)  Dibenz [a, h] anthracène (DB [a, h] A)

Cela signifie que la conformité technique des appareils utilisés pour la « mesure fixe » est jugée par le LCSQA avec l'aide des utilisateurs, après avis du Ministère en charge de l'environnement le cas échéant. Cela

  décembre 2004 concernant l'arsenic, le cadmium, le mercure, le nickel et les hydrocarbures aromatiques polycycliques dans l'air ambiant (dans les PM 10 ), -la Directive n°2008/50/CE du 21 mai 2008 concernant la qualité de l'air ambiant et un air 'agissant des méthodes dites « équivalentes aux méthodes de référence », il a été accepté le fait que les méthodes « équivalentes » correspondent à des appareils faisant l'objet d'une démonstration d'équivalence (sur la base du Guide européen de la Commission Européenne 3 ). Dans ce cas, le Ministère chargé de l'environnement accepte les appareils jugés équivalents sur la base de rapports d'essai effectués par les membres du LCSQA sur le territoire national ou délivrés dans d'autres Etats membres. aboutit à une liste d'appareils gérée par le LCSQA 4 .

	Les référentiels normatifs en vigueur à la date du présent document pour la majorité des polluants
	réglementés sont résumés en annexe II. La mise à jour par rapport aux éléments mentionnés dans
	les textes réglementaires en vigueur est disponible sur le site du LCSQA.

pur pour l'Europe (couvrant les polluants SO 2 , NO/NO 2 /NO x , PM 10 et PM 2,5 -en concentration massique -, le plomb -dans les PM 10 -, le benzène, le monoxyde de carbone, l'ozone -et ses précurseurs -et des composés spécifiques dans les PM 2,5 ), -la Directive n° 2015/1480 (UE) du 28 août 2015 modifiant plusieurs annexes des 2 directives précédentes établissant les règles concernant les méthodes de référence, la validation des données et l'emplacement des points de prélèvement pour l'évaluation de la qualité de l'air ambiant.

Les polluants concernés par le présent document sont ceux listés dans l'arrêté du 16/04/21 (Annexe I, points 1.1, 1.2.b et 1.2.c) et sont rappelés en annexe I. Dans ce cadre de surveillance réglementaire, la qualité de l'air ambiant doit être évaluée en utilisant des techniques spécifiquement désignées respectant des objectifs de qualité dépendant du niveau de polluant visé. La « mesure fixe » désigne la technique de mesure jugée la plus performante car répondant aux objectifs de qualité de données les plus ambitieux (cf. annexe 5 de l'arrêté du 16/04/21). Lorsque les objectifs de qualité de données sont moins élevés (ou en l'absence de seuil réglementaire spécifique -ex : le mercure, les précurseurs d'ozone, les composés spécifiques dans les PM 2.5 ), des méthodologies jugées moins contraignantes peuvent éventuellement être utilisées, comme la « mesure indicative », la « modélisation » ou l'« estimation objective », ou encore des combinaisons entre ces différentes méthodes. Il est stipulé dans l'article 16 de l'arrêté du 16/04/21 que : « L'AASQA met en oeuvre les démarches d'assurance de la qualité suivantes : 1 Arrêté du 29 juillet 2010 portant désignation d'un organisme chargé de la coordination technique de la surveillance de la qualité de l'air au titre du code de l'environnement (livre II, titre II) -l'utilisation pour les mesures fixes des méthodes de référence définies à l'annexe 6.1. Des méthodes équivalentes aux méthodes de référence peuvent être utilisées, après approbation par le ministère chargé de l'environnement, sur avis du LCSQA. Ces méthodes sont précisées dans le référentiel technique national 2 ; -l'utilisation, pour la réalisation de ses missions réglementaires, des équipements bénéficiant d'un avis de conformité technique du LCSQA ». S

Le demandeur de constat de conformité technique

  et des spécifications techniques associées le cas échéant) sur la liste des appareils déclarés conformes gérée par le LCSQA, -suivi de l'évolution du statut d'appareil conforme en cas de modification technique de l'appareillage et mise à jour de la liste des appareils déclarés conformes le cas échéant. Les membres du LCSQA évaluent les éléments techniques du dossier ainsi que les résultats de tests, en collaboration avec la Commission de Suivi « Suivi Technique des AASQA » et, le cas échéant, toute autre Commission de Suivi en fonction des besoins.  des informations communiquées par les AASQA (retour d'expérience d'utilisateurs), notamment via la Commission de Suivi « Suivi Technique des AASQA » et, le cas échéant, toute autre Commission de Suivi en fonction des besoins ;  des éléments fournis par le demandeur de vérification de conformité en cas de modification technique de son appareillage (cf. § 3.5). Les membres du LCSQA analysent les renseignements ainsi collectés, en collaboration avec la Commission de Suivi « Suivi Technique des AASQA » et, le cas échéant, toute autre Commission de Suivi en fonction des besoins ainsi qu'avec le demandeur de constat de conformité. Lorsque l'analyse de l'ensemble des informations fait apparaître un problème persistant sur la qualité ou la transmission des mesures d'un appareillage bénéficiant du statut de conformité technique, un nouvel avis technique est émis par le LCSQA et remis aux participants au processus pour information. En relation avec le Ministère en charge de l'environnement, les membres du LCSQA assurent la diffusion de la liste des appareillages bénéficiant du constat de conformité technique et sa mise à jour régulière (à minima une fois par an, sur le portail internet du LCSQA).

	2.3 Il est du ressort de l'organisme souhaitant voir son appareillage octroyer un constat de conformité
	technique de contacter le LCSQA en vue de prendre connaissance des modalités du processus de
	vérification (éléments à fournir, échéancier...).
	Le processus n'est enclenché qu'à la remise par le demandeur de l'ensemble des éléments demandés
	(cf. 3.1).
	Il est recommandé que le système de management de la qualité du constructeur soit conforme aux
	exigences des normes NF EN ISO 9001 6 et NF EN 15267-1&2 7
	A partir du suivi technique assuré par les membres du LCSQA, l'évolution du statut de conformité
	technique d'un appareillage est possible selon les 3 cas suivants :
	-maintien (voire extension) ;
	-suspension temporaire ;
	-suspension définitive.

Note : La vérification de la conformité technique peut inclure des tests en laboratoire et/ou sur le terrain, dans le cas où les conditions d'essai décrites dans le dossier ne sont pas représentatives des conditions usuelles rencontrées sur le territoire français (conditions environnementales telles que la gamme de concentration en polluant, modalités techniques d'utilisation telles qu'en station fixe climatisée ou en cabine mobile individuelle). Les tests complémentaires sur le terrain viseront à couvrir au moins une période de mesure estivale et hivernale.

Les membres du LCSQA sont destinataires :

 des informations fournies par le demandeur sur la maîtrise de la qualité de la fabrication, de sorte que tous les appareillages bénéficiant du constat de conformité technique continuent de respecter les critères de performance spécifiés dans le dossier initial ; Note : les informations appropriées peuvent être d'ordre métrologique définies dans un référentiel spécifique (ex : norme), d'ordre organisationnel (ex : qualité du SAV), de nature financière (ex : coût de pièces détachées indispensables) ou concerner le transfert de données (communication avec le Système d'Acquisition de Mesure). Dans ce dernier cas, elles peuvent être alors spécifiées dans un cahier des charges technique dans le cadre d'un marché négocié entre le fournisseur et les AASQA. Note : Il est rappelé qu'un constat de conformité technique est attribué à un matériel au nom de l'organisation ayant effectué la demande initiale. Dans le cas d'un distributeur, la conformité vis-àvis des exigences de la norme NF EN ISO 9001 est recommandée. De plus, en tant que représentant du constructeur, le distributeur doit avoir son accord écrit pour toute action dans le cadre du processus.

  Les AASQA font bénéficier au processus de leur retour d'expérience sur les appareillages bénéficiant d'un constat de conformité technique ou concernés par une demande, par exemple au travers d'essais documentés dans leurs conditions usuelles de mesure. Ce retour peut s'effectuer par le biais de la Commission de Suivi « Suivi Technique des AASQA » et, le cas échéant, toute autre Commission de Suivi en fonction des besoins, par communication de rapports d'essais aux membres du LCSQA ou par l'intermédiaire de questionnaires dédiés. Ces informations sont analysées par les membres du LCSQA afin d'assurer le suivi du parc instrumental français. En cas de problème persistant sur la qualité ou la transmission des mesures d'un appareillage bénéficiant du constat de conformité technique, ces informations contribuent le cas échéant à un nouvel avis technique élaboré par le LCSQA (cf. § 2.2).

10 ans est requise. Il convient que le demandeur indique le délai minimum d'intervention concernant la fourniture en pièces détachées, l'intervention pour résolution des dysfonctionnements et la reprise d'appareil en fin de vie.

Suite au verdict lors du processus initial (ou dans le cadre du suivi ultérieur), le demandeur peut, le cas échéant, émettre un avis auprès du LCSQA. Le demandeur doit signaler au LCSQA toute modification de l'appareillage bénéficiant du constat de conformité technique (cf. § 3.5). Lorsque des essais complémentaires sont nécessaires (initialement ou lors du suivi de conformité), le demandeur doit pouvoir mettre à disposition des membres du LCSQA un nombre d'appareils suffisant par rapport aux essais complémentaires demandés (de 1 à 5 au maximum). Dans le cas d'un appareil équivalent à la méthode de référence, l'incertitude déterminée lors de la Démonstration d'Equivalence et des conditions d'essais associés conditionne ce nombre. La mise à disposition d'appareils doit être assurée sur la période de commercialisation de l'appareil. 6 NF EN ISO 9001 « Systèmes de management de la qualité -Exigences » 7 NF EN 15267-1 « Qualité de l'air -Certification des systèmes de mesurage automatisés -Partie 1 : Principes généraux » et NF EN 15267-2 « Qualité de l'air -Certification des systèmes de mesurage automatisés -Partie 2 : Évaluation initiale du système de gestion de la qualité des fabricants d'AMS et surveillance après certification du procédé de fabrication » 2.4 Les AASQA

  soft ou sur le plan technique) doit être justifiée par le constructeur et doit être documentée.  l'engagement par écrit d'une mise à disposition d'appareils pour les membres du LCSQA pour le suivi de la conformité technique (notamment dans le cas d'appareils dits « équivalents à la méthode de référence ») ;  tout document jugé utile par le demandeur ou par les membres du LCSQA permettant de documenter un retour d'expérience sur l'appareil (article scientifique, rapport d'étude, rapport d'activités, détail des ventes dans d'autres pays -UE ou hors UE -, liste des appareils liés au demandeur bénéficiant déjà en France de constat de conformité technique, etc...).

  Le processus général débute à partir d'une demande de constat de conformité technique auprès de la direction exécutive du LCSQA et de la remise du dossier initial complet (sous format électronique) par le demandeur au LCSQA. Les coordonnées des interlocuteurs au LCSQA sont précisées en Annexe III. Sous réserve de la conformité du dossier et des résultats d'essais complémentaires estimés nécessaires par les membres du LCSQA (en collaboration avec la Commission de Suivi « Suivi Technique des AASQA », un avis technique est communiqué au CPS pour avis. : les points  et  peuvent amener le LCSQA, avec l'accord du ministère en charge de l'environnement, à ne pas accepter de nouvelles demandes de la part du demandeur pour d'autres matériels, en fonction de la cause de suspension et tant que celle-ci n'est pas levée.

	La conclusion sur la conformité technique de l'appareillage suite au CPS est communiquée par le
	LCSQA au demandeur ainsi qu'à l'ensemble du dispositif national de surveillance de la qualité de l'air
	(notamment pour une prise en compte dans la stratégie d'équipement).
	Une fois l'octroi du constat de conformité technique par un appareillage, son suivi est assuré par le
	LCSQA (cf. 2.2).
	En fonction des conclusions du suivi de constat de conformité technique (ex : résultats du suivi
	d'équivalence) ou en cas de modifications apportées sur l'appareillage concerné (cf. 3.5), un nouveau
	constat est émis par les membres du LCSQA.
	Ce nouveau constat peut être :
	1. maintenu à l'identique (c'est-à-dire inchangé) ;
	2. maintenu avec changements (extensions ou restrictions : par exemple, changement de
	configuration technique, modification du contexte d'utilisation…) ;
	3. suspendu temporairement (jusqu'à la levée de la cause de suspension) ;
	4. suspendu définitivement.

Note

  Une influence significative augmente ou réduit les performances de l'appareillage bénéficiant du constat de conformité par rapport aux caractéristiques de performance obtenues lors de la demande initiale. Le demandeur doit évaluer toutes les modifications techniques apportées à l'appareillage bénéficiant du constat de conformité. Lorsque des modifications de conception de type 1 ou 2 ont été effectuées, le demandeur doit alors en informer le LCSQA qui les étudiera en collaboration avec les membres du LCSQA et la Commission de Suivi « Suivi Technique des AASQA ». Dans l'attente de la conclusion, le constat de conformité technique obtenu initialement n'est pas étendu aux nouveaux appareils. Dans le cas de modification de type 1, le demandeur doit fournir au LCSQA les éléments démontrant que cette modification n'a pas d'influence significative (ex : résultats d'essais). En fonction des éléments fournis, les membres du LCSQA peuvent devoir effectuer des tests complémentaires (par exemple lorsque les conditions des essais diffèrent des conditions usuelles rencontrées sur le territoire français). Le résultat de ces tests complémentaires peut impacter le constat de conformité technique initial de l'appareillage (cf. § 3.2). Lorsque des modifications de type 2 sont identifiées, une évaluation complémentaire est menée par les membres du LCSQA, en collaboration avec les participants au processus. Cette évaluation complémentaire peut amener les membres du LCSQA à devoir effectuer des tests complémentaires. Le résultat de cette évaluation complémentaire peut impacter le constat de conformité technique initial de l'appareillage (cf. § 3.2).

	Les 3 classes suivantes de modification pouvant être effectuée sur un appareillage bénéficiant du
	constat de conformité sont les suivantes :
	 type 0 : modification n'ayant aucune influence mesurable sur les performances
	demandées ;
	 type 1 : modification susceptible d'avoir une influence sur les performances demandées
	mais des essais ont pu démontrer que cette modification n'a pas d'influence significative ;
	 type 2 : modification ayant une influence significative sur les performances
	demandées.

Téléchargeable sur https://www.lcsqa.org/fr/referentiel-technique-national

Guide to the demonstration of equivalence of ambient air monitoring methods -Report by an EC Working Group on Guidance for the Demonstration ofEquivalence (2010)

Téléchargeable sur https://www.lcsqa.org/fr/conformite-technique-appareils-mesure

Guide national « Conception, implantation et suivi des stations françaises de surveillance de la qualité de l'air » (version en vigueur téléchargeable sur https://www.lcsqa.org/fr/referentiel-technique-national)

disponibilité de l'appareillage : fraction du temps total pendant laquelle chaque appareillage individuel sujet à demande de constat fournit des données de mesure exploitables relatives à la concentration de l'air ambiant.

période de fonctionnement sans surveillance : intervalle d'entretien, c'est-à-dire période de temps la plus longue sans intervention recommandée par le demandeur sur chaque appareillage sujet à demande de constat de conformité.


Marco Antonio Chiminazzo
email: marcochiminazzo@gmail.com

Aline Bertolosi Bombo

Tristan Charles-Dominique

Alessandra Fidelis

To protect or to hide: Why not both? An investigation of fire-related strategies in Cerrado woody species

Keywords: Belowground organs, bark production, bark thickness, clonal growth, resprouting, underground storage organs

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Species persistence in fire-prone ecosystems is highly dependent on their ability to survive fire. During a fire event, not all parts of the plants are equally exposed to the flames, with different traits being required to promote fire-survival [START_REF] Chiminazzo | Your best buds are worth protecting: Variation in bud protection in a fire-prone cerrado system[END_REF][START_REF] Scalon | Diversity of functional trade-offs enhances survival after fire in Neotropical savanna species[END_REF][START_REF] Wigley | A handbook for the standardised sampling of plant functional traits in disturbance-prone ecosystems, with a focus on open ecosystems[END_REF]. This is the consequence of the distribution of aboveground structures that are exposed or not to the flames: higher structures such as branches and twigs can be positioned outside the flames and are only exposed to the flame plume. On the other end, older parts of the trunk and/or main stems are located closer to the ground and are thus exposed to flames [START_REF] Chiminazzo | Your best buds are worth protecting: Variation in bud protection in a fire-prone cerrado system[END_REF][START_REF] Dantas | The lanky and the corky: fire-escape strategies in savanna woody species[END_REF][START_REF] Gignoux | Alternative fire resistance strategies in savanna trees[END_REF][START_REF] Graves | Outer bark thickness decreases more with height on stems of fire-resistant than fire-sensitive Floridian oaks (Quercus spp.; Fagaceae)[END_REF]. Plant structures can be positioned belowground and be well insulated by the soil, where they barely suffer from fire damagea clear contrast with aboveground structures located within the flame zone that are often heavily damaged by the flames [START_REF] Choczynska | A soil heat and water transfer model to predict belowground grass rhizome bud death in a grass fire: Belowground grass rhizome bud death in a fire[END_REF][START_REF] Hoffmann | The role of topkill in the differential response of savanna woody species to fire[END_REF][START_REF] Kavanagh | A Way Forward for Fire-Caused Tree Mortality Prediction: Modeling A Physiological Consequence of Fire[END_REF][START_REF] Pausas | Unearthing belowground bud banks in fire-prone ecosystems[END_REF]. Consequently, how plants experience fires is strongly dependent on their allocation to above-and/or perennial organs positioned belowground.

Many plant species from fire-prone ecosystems have traits that allow them to resprout and persist after fire [START_REF] Bond | Ecology of sprouting in woody plants: the persistence niche[END_REF], such as the development of new branches mostly from buds located above-and/or belowground [START_REF] Burrows | Epicormic strand structure in Angophora, Eucalyptus and Lophostemon (Myrtaceae) -implications for fire resistance and recovery[END_REF][START_REF] Clarke | Resprouting as a key functional trait: how buds, protection and resources drive persistence after fire[END_REF][START_REF] Klimešová | Bud banks and their role in vegetative regeneration -A literature review and proposal for simple classification and assessment[END_REF]. While belowground resprouting is very common [START_REF] Pausas | Towards understanding resprouting at the global scale[END_REF]Pilon et al., 2021;[START_REF] Zupo | Post-fire regeneration strategies in a frequently burned Cerrado community[END_REF], aboveground resprouting usually occur high in the canopy most often in tall trees [START_REF] Chiminazzo | Your best buds are worth protecting: Variation in bud protection in a fire-prone cerrado system[END_REF][START_REF] Scalon | Diversity of functional trade-offs enhances survival after fire in Neotropical savanna species[END_REF][START_REF] Souchie | Post-fire resprouting strategies of woody vegetation in the Brazilian savanna[END_REF]. Species with this latter strategy usually pay a high cost to protect their trunk (e.g., by producing considerable amounts of bark; [START_REF] Pausas | Bark thickness and fire regime[END_REF][START_REF] Rosell | Bark functional ecology: evidence for trade-offs, functional coordination, and environment producing bark diversity[END_REF], but gain the ability to maintain a large storage compartment aboveground, allowing to display new foliage in higher strata when compared to species that have resprouted from the ground [START_REF] Crisp | Flammable biomes dominated by eucalypts originated at the Cretaceous-Palaeogene boundary[END_REF]. On the other hand, species with a strategy based on basal and belowground resprouting are assumed to minimize costs of protecting their aerial parts: they rely on both the maintenance of a viable bud bank at the plant base and belowground through the development of perennating organs that can store reserve either at the soil surface or belowground [START_REF] Pausas | Unearthing belowground bud banks in fire-prone ecosystems[END_REF].

Aboveground, bark plays a key role in woody plants for protecting their structures from fire. The bark creates a protective layer that surrounds vital inner tissues and insulates them from air, fire, cold, and pathogens [START_REF] Burrows | Buds buried in bark: the reason why Quercus suber (cork oak) is an excellent post-fire epicormic resprouter[END_REF][START_REF] De Antonio | The role of bud protection and bark density in frost resistance of savanna trees[END_REF][START_REF] Gashaw | Post-fire regeneration strategies and tree bark resistance to heating in frequently burning tropical savanna woodlands and grasslands in Ethiopia[END_REF][START_REF] Lawes | Costs and benefits of relative bark thickness in relation to fire damage: a savanna/forest contrast[END_REF][START_REF] Rosell | Inner bark as a crucial tissue for non-structural carbohydrate storage across three tropical woody plant communities[END_REF].

During a fire event, bark notably prevents the heat from killing the cambium and inducing xylem deformation [START_REF] Gashaw | Post-fire regeneration strategies and tree bark resistance to heating in frequently burning tropical savanna woodlands and grasslands in Ethiopia[END_REF][START_REF] Hacke | Trends in wood density and structure are linked to prevention of xylem implosion by negative pressure[END_REF][START_REF] Lawes | Costs and benefits of relative bark thickness in relation to fire damage: a savanna/forest contrast[END_REF][START_REF] Lawes | Bark thickness determines fire resistance from fire-prone tropical savanna in north Australia[END_REF][START_REF] Michaletz | Moving beyond the cambium necrosis hypothesis of post-fire tree mortality: cavitation and deformation of xylem in forest fires[END_REF]; it also protects buds located below its surface allowing future resprouting [START_REF] Bond | Ecology of sprouting in woody plants: the persistence niche[END_REF][START_REF] Burrows | Epicormic strand structure in Angophora, Eucalyptus and Lophostemon (Myrtaceae) -implications for fire resistance and recovery[END_REF][START_REF] Charles-Dominique | Bud protection: a key trait for species sorting in a forest-savanna mosaic[END_REF][START_REF] Chiminazzo | Your best buds are worth protecting: Variation in bud protection in a fire-prone cerrado system[END_REF]. In ecosystems with fires fueled by grasses (e.g., savannas), fires are of low intensity and high frequency [START_REF] Archibald | Biological and geophysical feedbacks with fire in the Earth system[END_REF][START_REF] Bond | Beyond the forest edge: Ecology, diversity and conservation of the grassy biomes[END_REF]. Hence, in savannas, the main survival strategy of woody species is to produce and accumulate bark fast enough (without bark shedding) before the next fire event to protect their aboveground parts, instead of accumulating a large amount of bark over a long period [START_REF] Charles-Dominique | Fire frequency filters species by bark traits in a savanna-forest mosaic[END_REF]. Bark is such an important trait for species from fire-prone ecosystem that it can predict community assembly across fire-sensitive forests and fire-prone savannas based on how much bark woody species produce (at least 0.13 mm of bark per growth unit in the Cerrado; Chiminazzo et al., 2023a; also see [START_REF] Charles-Dominique | Fire frequency filters species by bark traits in a savanna-forest mosaic[END_REF].

While bark is important for woody species to protect their aboveground organs, developing perennial belowground organs also accounts for the success of plant species during their development in fire-prone ecosystems [START_REF] Klimešová | Bud banks and their role in vegetative regeneration -A literature review and proposal for simple classification and assessment[END_REF][START_REF] Maurin | Savanna fire and the origins of the 'underground forests' of Africa[END_REF][START_REF] Ott | The ecology and significance of belowground bud banks in plants[END_REF]. These organs are related to different functions: in addition to bear buds that will assure post-disturbance resprouting, they can allow clonal growth through lateral spread (as in the case of woody rhizomes), assure a large part of resource storage, improve anchorage, and organize the fine roots involved in resource acquisition [START_REF] Archibald | Biological and geophysical feedbacks with fire in the Earth system[END_REF][START_REF] Bardgett | Going underground: root traits as drivers of ecosystem processes[END_REF][START_REF] Klimešová | Bud banks and their role in vegetative regeneration -A literature review and proposal for simple classification and assessment[END_REF][START_REF] Laliberté | Below-ground frontiers in trait-based plant ecology[END_REF][START_REF] Maurin | Savanna fire and the origins of the 'underground forests' of Africa[END_REF][START_REF] Ott | The ecology and significance of belowground bud banks in plants[END_REF]. In disturbance-driven ecosystems, the presence of budbearing underground storage organs (USOs) is a key trait assuring plant survival and persistence and accounting for vegetation resilience [START_REF] Bombo | Fire exclusion changes belowground bud bank and bud-bearing organ composition jeopardizing open savanna resilience[END_REF][START_REF] Pilon | The diversity of post-fire regeneration strategies in the cerrado ground layer[END_REF][START_REF] Zupo | Post-fire regeneration strategies in a frequently burned Cerrado community[END_REF], with belowground resprouting being 25 times more common than aerial resprouting in the Cerrado [START_REF] Chiminazzo | Your best buds are worth protecting: Variation in bud protection in a fire-prone cerrado system[END_REF]. Moreover, USOs are also important for species capable of resprouting aboveground as they allow resprouting to occur after the aerial structures were consumed or heavily damaged by the flames. Consequently, USOs allow plants' persistence in the post-fire ecosystem even after their aerial buds were damaged by the flames [START_REF] Charles-Dominique | Bud protection: a key trait for species sorting in a forest-savanna mosaic[END_REF][START_REF] Grady | Caught in a fire trap: Recurring fire creates stable size equilibria in woody resprouters[END_REF][START_REF] Souchie | Post-fire resprouting strategies of woody vegetation in the Brazilian savanna[END_REF]. However, even though there are several advantages for plants investing in belowground biomass, there are drawbacks when it comes to resprouting, since basal resprouting requires more time and resources to reconstitute an equivalent biomass and the plant's spatial occupation prior to the disturbance [START_REF] Clarke | Resprouting as a key functional trait: how buds, protection and resources drive persistence after fire[END_REF][START_REF] Crisp | Flammable biomes dominated by eucalypts originated at the Cretaceous-Palaeogene boundary[END_REF] Interestingly, few studies analyzed whether developing USOs trade-off with protecting aboveground structures or not. If having USOs does not preclude protecting the aboveground structures by positioning the buds deep under the bark layer and/or developing a thick bark [START_REF] Burrows | A wide diversity of epicormic structures is present in Myrtaceae species in the northern Australian savanna biome -implications for adaptation to fire[END_REF][START_REF] Charles-Dominique | Bud protection: a key trait for species sorting in a forest-savanna mosaic[END_REF][START_REF] Chiminazzo | Your best buds are worth protecting: Variation in bud protection in a fire-prone cerrado system[END_REF], an allocation trade-off is expected as both developing thick bark and large belowground organs incur high costs (e.g., [START_REF] Dantas | The lanky and the corky: fire-escape strategies in savanna woody species[END_REF][START_REF] Dantas | The lanky and the corky: fire-escape strategies in savanna woody species[END_REF][START_REF] Gignoux | Alternative fire resistance strategies in savanna trees[END_REF][START_REF] Lawes | Costs and benefits of relative bark thickness in relation to fire damage: a savanna/forest contrast[END_REF][START_REF] Pausas | Unearthing belowground bud banks in fire-prone ecosystems[END_REF]. Several woody plants have been reported for resprouting both below and aboveground buds after fire [START_REF] Charles-Dominique | Bud protection: a key trait for species sorting in a forest-savanna mosaic[END_REF][START_REF] Chiminazzo | Your best buds are worth protecting: Variation in bud protection in a fire-prone cerrado system[END_REF][START_REF] Scalon | Diversity of functional trade-offs enhances survival after fire in Neotropical savanna species[END_REF][START_REF] Souchie | Post-fire resprouting strategies of woody vegetation in the Brazilian savanna[END_REF], suggesting that under certain conditions, a strategy based on paying costs to both aboveground protection and belowground storage could emerge. Based on these observations, we ask: are species with underground storage organs also protecting their aboveground structures with bark? Does this protection vary according to the type and main function of these organs? For instance, would species with woody rhizomes capable of spreading laterally display different bark production/aerial bud protection when compared to those persisting on spot through organs like the xylopodium and the root crown?

Considering the importance of investing in aboveground and belowground perennial organs for species from fire-prone ecosystems, in this study we tested if i.

species with specialized bud-bearing belowground organs also produce enough bark to allow them to persist in fire-prone ecosystems in the Cerrado (at least 0.13 mm/growth season; Chiminazzo et al., 2023a) ii. bark production differs according to the type of specialization of the underground storage organ (lateral spreading vs on spot persistence)

We assessed the bark production, aboveground bud protection and potential height of 24 Cerrado woody species that develop common types of underground storage organs with different ecological specialization: woody rhizome (lateral spread),

xylopodium and root crown (on spot persistence).

Materials and Methods

Study area

The Cerrado (tropical savanna in Brazil) is composed of different vegetation types ranging from fire-prone savannas and grasslands to fire-sensitive forests [START_REF] Coutinho | O conceito de cerrado[END_REF][START_REF] Eiten | The Cerrado Vegetation of Brazil[END_REF]. Fire is closely related to the evolution of plant species of the Cerrado and imposes a strong filter promoting fire-adapted species in savannas [START_REF] Coutinho | Fire in the Ecology of the Brazilian Cerrado[END_REF]Dantas, Batalha, et al., 2013;[START_REF] Eiten | The Cerrado Vegetation of Brazil[END_REF][START_REF] Hoffmann | Ecological thresholds at the savannaforest boundary: how plant traits, resources and fire govern the distribution of tropical biomes[END_REF][START_REF] Simon | Recent assembly of the Cerrado, a neotropical plant diversity hotspot, by in situ evolution of adaptations to fire[END_REF]. For this study, we sampled species at the Santa Bárbara Ecological Station (SBES,22º48'59'' S,19º14'12'' W), in southeastern Brazil. This protected area is composed of different fire-prone savanna vegetation types [START_REF] Melo | Estação Ecológica de Santa Bárbara -Plano de Manejo[END_REF][START_REF] Ribeiro | As principais fitofisionomias do bioma Cerrado[END_REF], from campo sujo (open savanna) to cerrado sensu stricto (woody savanna), as well as fire-free vegetation types like the cerradão and the seasonally semideciduous forests. The vegetation at the SBES is exposed to the markedly seasonality of the Cerrado, experiencing hot and wet summers from October to April (mean annual temperature: 24 ºC, total rainfall: 1,000 mm) and dry and mild winters from May to September (mean annual temperature:

18.6 ºC, total rainfall: 400 mm; [START_REF] Melo | Estação Ecológica de Santa Bárbara -Plano de Manejo[END_REF].

Species selection and traits description

We compiled information from different sources to select Cerrado woody species i) occurring in fire-prone and open-canopy ecosystems; ii) with underground storage organ type described [START_REF] Pausas | Unearthing belowground bud banks in fire-prone ecosystems[END_REF][START_REF] Pilon | The diversity of post-fire regeneration strategies in the cerrado ground layer[END_REF]and personal observations); iii) and with information about their bark growth rate and aerial bud protection from our previous studies [START_REF] Chiminazzo | Your best buds are worth protecting: Variation in bud protection in a fire-prone cerrado system[END_REF]Chiminazzo et al., 2023a). In our previous studies, we sampled at least three individuals of each species depending on their presence in sampled plots across different savannas.

We standardized the nomenclature for underground storage organ types following [START_REF] Pausas | Unearthing belowground bud banks in fire-prone ecosystems[END_REF]. Following the criteria, we were able to describe 24 woody species (Table 1) with three different types of belowground bud-bearing organs [START_REF] Pausas | Unearthing belowground bud banks in fire-prone ecosystems[END_REF]: i) xylopodium, a woody basal swelling that usually origins from the hypocotyl and carries axillary or adventitious buds near the soil surface; ii) root crown, defined by the root-shoot transition zone that carries dormant axillary buds grouped in clusters above the soil surface, and iii) woody rhizome, a long woody stem distributed horizontally under the soil surface that carry buds mainly concentrated on the nodes (Figure 1). visible growth rings or by the number of well-developed growth units at the measured section of branches, selected to have an analogous morphology to the main stem [START_REF] Wigley | A handbook for the standardised sampling of plant functional traits in disturbance-prone ecosystems, with a focus on open ecosystems[END_REF]. Finally, we retrieved information from literature and herbaria about the maximal height of each species (except for Pouteria subcaerulea that was recorded visually in our study site).

Statistical analyses

We used generalized mixed effect models (GLMM) to model the variation in bark growth rate (BGR) depending on the type of underground storage organ (USO;

xylopodium, root crown, or woody rhizome). We considered the BGR as a response variable, the type of USO as a predictor variable, and the species and their individuals as a random effect. We also used GLMM to model how BGR changes across different species displaying the same type of USO, and across every species regardless of the type of USO. In these two cases, we considered the species as a fixed effect and their individuals as a random effect. After testing the significance of the models, we performed pairwise comparisons between each type of USO and each species displaying the same type of USO. Due to multiple testing across contrasts in these cases, we adjusted the p-values by using the Bonferroni correction method to avoid the type 1 error. The same approach was used to test differences in bark production depending on the specialization of each type of organ (lateral spread or on spot persistence).

After checking that BGR indeed varied depending on the type of USO, we performed a principal component analysis (PCA) to visualize if changes in the BGR would be related with each species aerial bud protection (IBP), their maximum height, and their belowground functional syndromes (persisting fire on spot or spreading laterally). The PCA was carried out using the factoextra (v 1.0.7) package, while GLMMs were modeled and tested using the lme4 (v 1.1.29), lsmeans (v 2.30.0), and emmeans (v. 1.7.3) packages. All analyses were performed in the R environment using the R software (v. 4.2.0; R Core Team, 2023).

Results

Bark production varied from 0.06 to 0.90 mm/growth unit among all species (Table 1). Bark growth rate differed among the species (P < 0.001) and among species with the same type of belowground organ (P < 0.001). Mean bark production differed across the three types of belowground organs (P < 0.001, Table 2). Species with xylopodium showed a mean bark production of 0.23±0.17 mm, while species with root crown had mean bark production of 0.36±0.25 mm and species with woody rhizome displayed bark production of 0.57±0.23 mm/growth unit. A significant difference was detected across organs with different ecological specialization, with bark growth rate and aerial bud protection differing between species with woody rhizome (lateral spread) and xylopodium (P < 0.001) and root crown (P < 0.001, both on spot persistence) but not between the xylopodium and the root crown organs.

Lastly, the ordination of species concerning their growth strategy (on spot or spreading laterally) revealed a clear division explained by their bark growth rate, their aerial bud protection, and the species potential height, with the principal components explaining 92,4% of all data variation (Figure 3).

[Table 1 [START_REF] Chiminazzo | Your best buds are worth protecting: Variation in bud protection in a fire-prone cerrado system[END_REF][START_REF] Wigley | A handbook for the standardised sampling of plant functional traits in disturbance-prone ecosystems, with a focus on open ecosystems[END_REF]. The red dashed line indicates the bark threshold separating species from fire-sensitive to fire-prone ecosystems in the Cerrado (0.13 mm; Chiminazzo et al., 2023a). This threshold relates to the minimal amount of bark produced by woody communities capable of persisting in fire-prone ecosystems. Species names are colored according to their predominant growth form

Discussion

The results of this study reveal that escaping fire belowground and resisting the flames aboveground are strategies that can occur mutually across different types of underground storage organs (USOs). We demonstrate that the ecological specialization of the belowground organ matters for understanding the strategies being adopted by these plants, as species persisting on spot showed less aboveground protection than those spreading laterally and displaying higher potential height. Consequently, these observations raise further questions: why protect stems from fire when these species are already safe from fire damage? And if there is no major constraint to do so, why do not all plants perform both aerial and belowground protection? In the following we discuss what these differences mean for species from fire-prone ecosystems in the Cerrado and why plants displaying both below and aboveground protection reveal that species are highly adapted to fire.

Bark production differs among species displaying the same type of underground storage organ. An impressive variation of bark growth rate was found among species, varying from 0.06 and 0.90 mm/growing unit, which should translate into different levels of protection of the aboveground tissues against fire (Charles- et al., 2017;[START_REF] Pellegrini | Convergence of bark investment according to fire and climate structures ecosystem vulnerability to future change[END_REF]. Interestingly, each USO type comprised species with their bark production distributed all along the gradient from low to high levels of bark protection aboveground. This suggests that species can allocate resources to both develop thick bark aboveground and invest in bud-bearing underground storage organsat least in the Cerrado, where fire is the main disturbance in open ecosystems [START_REF] Dantas | Disturbance maintains alternative biome states[END_REF][START_REF] Hoffmann | Ecological thresholds at the savannaforest boundary: how plant traits, resources and fire govern the distribution of tropical biomes[END_REF]. We suggest that by doing so, species can maintain both an above-and belowground viable bud bank, being able to resprout whichever situation is met: if fires are less severe, aboveground resprouting may occur, while if fires are more severe and plants are top-killed, their persistence will be assured by resprouting from the belowground/basal bud bank [START_REF] Chiminazzo | Your best buds are worth protecting: Variation in bud protection in a fire-prone cerrado system[END_REF][START_REF] Clarke | Resprouting as a key functional trait: how buds, protection and resources drive persistence after fire[END_REF], being very difficult to kill a savanna woody species in the Cerrado Species with a xylopodium (on spot persistence) showed overall a lower bark production when compared to species that developed woody rhizomes (lateral spreading). Root crown species had both low and high bark production and intermediate levels of aerial bud protection (their bud protection was intermediate between species with xylopodium and species with woody rhizomes). Each of the USO type has different level of protection from fire [START_REF] Pausas | Unearthing belowground bud banks in fire-prone ecosystems[END_REF]: buds are exposed to flames in species with root crowns as they are directly located at the soil surface; by comparison, basal buds of xylopodium are protected from flames located below the soil surface (up to 10 cm deep). For species with root crown, having extra protection aboveground is therefore advantageous, while species with xylopodium have greater chance to resprout after fire since their buds are better insulated from the flames.

Dominique

Together with bark production, aerial bud protection by bark also differed among species displaying lateral spread and those persisting on spot. In this study, most Cerrado species sampled with woody rhizomes (lateral spread) are singlestemmed trees or treelets, different from species with xylopodium or root crown that often grow as multi-stemmed shrubs. Species with rhizomes had the greatest potential height. The need for thick bark and better bud protection is probably higher in woody rhizome that have a notably smaller bud bank than species with xylopodium and root crown [START_REF] Bombo | Fire exclusion changes belowground bud bank and bud-bearing organ composition jeopardizing open savanna resilience[END_REF]; conversely, species with xylopodium and root crown have a greater chance of resprouting from belowground which in turn help explaining their multi-stemmed architecture (e.g., [START_REF] Götmark | Why Be a Shrub? A Basic Model and Hypotheses for the Adaptive Values of a Common Growth Form[END_REF][START_REF] Scheffer | Why trees and shrubs but rarely trubs?[END_REF]Chiminazzo et al., 2023b). On the other hand, likely compensating for the smaller bud bank, species with woody rhizomes showed to better protect their aboveground buds with bark and position them outside the flame zone. We advocate that this strategy is beneficial for species with woody rhizomes, since they can colonize a greater space in the ecosystem [START_REF] Klimešová | Horizontal growth: An overlooked dimension in plant trait space[END_REF] while maintaining their branches and stems better protected from the flames in taller strata [START_REF] Chiminazzo | Your best buds are worth protecting: Variation in bud protection in a fire-prone cerrado system[END_REF][START_REF] Crisp | Flammable biomes dominated by eucalypts originated at the Cretaceous-Palaeogene boundary[END_REF]. These observations combined with their greater potential height suggest that these species are more likely to eventually escape the firetrap (see [START_REF] Bond | Fire and plants[END_REF].

Although Cerrado woody plants can produce great amounts of bark, investing in bark aboveground does not necessarily mean branch capacity of surviving fire.

This capacity relies on different traits that confer thermal insulation during fire events, like branch and bark density, inner bark proportion in relation to the outer bark, bark water content, and stem diameter [START_REF] Hacke | Trends in wood density and structure are linked to prevention of xylem implosion by negative pressure[END_REF][START_REF] Hoffmann | Ecological thresholds at the savannaforest boundary: how plant traits, resources and fire govern the distribution of tropical biomes[END_REF][START_REF] Kavanagh | A Way Forward for Fire-Caused Tree Mortality Prediction: Modeling A Physiological Consequence of Fire[END_REF][START_REF] Lawes | Bark thickness determines fire resistance from fire-prone tropical savanna in north Australia[END_REF][START_REF] Loram-Lourenço | A Structure Shaped by Fire, but Also Water: Ecological Consequences of the Variability in Bark Properties Across 31 Species From the Brazilian Cerrado[END_REF][START_REF] Pausas | Bark thickness and fire regime[END_REF][START_REF] Rosell | Inner bark as a crucial tissue for non-structural carbohydrate storage across three tropical woody plant communities[END_REF][START_REF] Rosell | Bark functional ecology: evidence for trade-offs, functional coordination, and environment producing bark diversity[END_REF]Scalon et al., 2021). According to Chiminazzo et al. (2023), a minimum of 0.13 mm of bark production is enough to differentiate species from fire-prone and fire-sensitive ecosystems in the Cerrado. However, this does not indicate that all species producing bark above the threshold are able to have their aerial parts well-protected against fire: some species produce bark above the threshold, but their stems rarely reach enough diameter to survive fire (e.g., Chromolaena squalida, Lippia origanoides)thus often being top killed and resprouting from belowground. Therefore, further studies should address experimental approaches combining bark production with plant architecture and allometry to better disentangle bark production from growing and resprouting patterns.

Finally, the results of this study highlight the importance of considering belowground traits when studying plant functional differentiation (e.g., [START_REF] Bardgett | Going underground: root traits as drivers of ecosystem processes[END_REF][START_REF] Klimešová | Horizontal growth: An overlooked dimension in plant trait space[END_REF][START_REF] Laliberté | Below-ground frontiers in trait-based plant ecology[END_REF][START_REF] Ott | The ecology and significance of belowground bud banks in plants[END_REF]. Taking into account the morphology of belowground organs (i.e., where and how the buds are protected) together with aboveground traits (here, bark production and aerial bud protection),

showed to relate with the type of growing strategy that plants display in the Cerrado fire-prone ecosystems (persisting on spot or spreading laterally) and to how much of biomass they invest aboveground through their potential height. Although trade-offs are often expected concerning alternative strategies, e.g., either escaping or resisting the flames [START_REF] Dantas | The lanky and the corky: fire-escape strategies in savanna woody species[END_REF], our results suggest that species combining several fire-survival strategies might also be selected in fire-prone ecosystems.

Conclusions

The great variation in rates of bark production aboveground throughout different types of bud-bearing underground storage organs indicate that Cerrado woody species invest in strategies that confer both above-and belowground protection against fire, allowing them to persist both if fires are more or less severe. Therefore, escaping the 'firetrap' can be combined with resisting the flame action, as Cerrado species are both fire-resisting aboveground (through investments in bark and aerial bud protection) and fire-escaping belowground (by hiding buds in organs belowground). Further studies should quantify which strategies are being promoted under different fire regimes and how they may impact vegetation feedbacks.

Funding

MAC received financial support from FAPESP (2018/21300-6, 2021/09269-9), ABB received financial support from FAPESP (2017/02934-1) and CNPq (150307/2022-5)

and AF receives grant from CNPq (312689/2021-7). This study received financial support from CNPq (406505/2018-7), FAPESP (2015/06743-0) and National Geographic Society (NGS 519036-18).

Author contributions

AF conceived and designed the study. MAC wrote the original draft and led the writing of the manuscript. MAC, ABB, and AF performed field sampling. MAC and TC-D analyzed the data. All authors participated actively in the execution of the study and gave final approval for publication.

Figure captions

Figure 1

 1 Figure 1 Types of belowground bud-bearing organs. a) Xylopodium associated with

Figure 2 :

 2 Figure 2: Bark growth rate according to the species and their belowground organ

(

  blue = shrubs, brown = trees). USOs illustrations were adapted from Pausas et al. (2018).

Figure 3 :

 3 Figure 3: Distribution of species with different types of aboveground strategies (on

Figure 1

 1 Figure 1 Types of belowground bud-bearing organs. a) Xylopodium associated with

Figure 2 :

 2 Figure 2: Bark growth rate according to the species and their belowground organ

Figure 3 :

 3 Figure 3: Distribution of species with different types of aboveground strategies (on

should come here. It does not fit the document with the numbered lines, so it was submitted as a separate .doc file] Table

		Bark production
	Predictors	Estimates	CI	p
	Intercept	0.37	0.20 -0.53	<0.001
	Woody rhizome	0.21	-0.01 -0.44	0.065
	Xylopodium	-0.13	-0.35 --0.09	0.244
	Marginal R²/ Conditional R²	0.269 / 0.896

2 Relationship between underground storage organs (predictors) and bark production. Estimates, confidence interval (CI), and their respective p-values (p) are shown, as well as marginal and conditional R² values of the model.

confidence level of 95% of the centroid calculated using the mean values of the 635 principal components for each category (lateral spread or on spot persistence).


F Chalon

F Montheillet

THE INTERACTION OF TWO SPHERICAL GAS BUBBLES IN AN INFINITE ELASTIC SOLID

Keywords: nuclear fuel, aging, bubbles, stress concentration, equivalent inclusion method

The elastic strain and stress fields between two bubbles of different sizes and different pressures were estimated by using the fundamental result of Eshelby. The equivalent inclusion method was extended to the case of two inclusions in an infinite elastic solid. This approach, which remains totally analytical, was compared successfully to finite element calculations. The mean stress provides information about gas diffusion between the bubbles: according to the results, the bubbles are likely to progressively equalize their sizes. Moreover, the derivation of the von Mises equivalent stress showed that its value, in the vicinity of the bubbles, is larger than the elasticity limit. Therefore, for a complete mechanical description of the problem, plasticity should be taken into account. In spite of its simplicity, this method nevertheless leads to results, which are very close to the prediction of numerical calculations.

Introduction

The elastic interaction of two spherical cavities with internal pressure (bubbles) is important from the standpoint of materials engineering and has already been investigated by several authors. One of the common applications is the aging of nuclear fuels: nuclear fission produces helium atoms, which form bubbles that eventually damage the material (Lässer [START_REF] Lässer | Tritium and Helium-3 in metals[END_REF]).

The knowledge of the stress field between two bubbles gives information about their further evolutions. More specifically, the von Mises equivalent stress level is related to the possible occurrence of local damage associated with plastic strain, while the mean stress (hydrostatic pressure) is associated with bubble growth. An analytical or/and numerical approach on this kind of material is necessary, since it is very difficult to carry out direct measurements of the material evolution with time. It is also relevant to derive analytical expressions of the mechanical interaction between two bubbles, that could be simply implemented into more complex numerical models. Sternberg and Sadowsky [START_REF] Sternberg | On the axisymmetric problem of the theory of elasticity for an infinite region containing two spherical cavities[END_REF] were the first to be interested in the problem of interaction of two spherical cavities of the same size. They solved it for a uniform field of tension at infinity using the Boussinesq [START_REF] Boussinesq | Applications des potentiels[END_REF] stress-function approach to obtain a series expansion solution. Other authors then extended this method to more than two cavities and for cavities of different sizes under uniaxial loading along the common axis of the cavities, Miyamoto [START_REF] Miyamoto | On the problem of theory of elasticity for a region containing more than two spherical cavities[END_REF], and uniaxial tension in the direction perpendicular to the axis of the cavities, Tsuchida, Nakahara and Kodama [START_REF] Tsuchida | On the asymmetric problem of elasticity theory for infinite elastic solid containing some spherical cavities[END_REF]. Shelley and Yu [START_REF] Shelley | The effect of two rigid spherical inclusions on the stresses in an infinite elastic solid[END_REF], still following Sternberg and Sadowsky [START_REF] Sternberg | On the axisymmetric problem of the theory of elasticity for an infinite region containing two spherical cavities[END_REF], developed this method for inclusions. Chen and Acrivos [START_REF] Chen | The solution of the equations of linear elasticity for an infinite region containing spherical inclusions[END_REF] extended it to an arbitrary strain field applied at infinity. Willis and Bullough [START_REF] Willis | The interaction of finite gas bubbles in a solid[END_REF] proposed a slightly different approach: they considered the total energy of the bubbles, i.e. the elastic energy, the energy of the gas, and the surface energy. Their approach was therefore not only mechanical but also thermodynamical. They gave a solution for the energy interaction between two excess pressure bubbles but they did not derive precisely the stress field from their results.

All the above authors, except Willis and Bullough, started their analysis with Boussinesq [START_REF] Boussinesq | Applications des potentiels[END_REF], Papkovitch [START_REF] Papkovitch | Solution générale des équations différentielles fondamentales d'élasticité, exprimées par trois fonctions harmoniques[END_REF] or Neuber [START_REF] Neuber | Kerbspannungslehre[END_REF] functions and gave series expansion solutions.

In this work, a different approach is proposed, based on the fundamental result of Eshelby [START_REF] Eshelby | The determination of the elastic field of an ellipsoidal inclusion and related problems[END_REF] extended to two inhomogeneities. The determination of the eigenstrain of each inclusion from the equivalent inclusion method is used to derive the stress and strain fields between the inclusions. The present approach involving cavities with an internal pressure is the same as for cavities in a material submitted to external loading, but it seems important to explain the transposition from one problem to the other. This method is quite general since it allows the interaction between two gas bubbles of different sizes and different pressures to be investigated. Three configurations are analysed, i.e. two identical bubbles with the same pressure, two bubbles of different sizes with identical pressures and two identical bubbles with different pressures. For each case, the mean stress and von Mises equivalent stress are displayed along the symmetry axis of the bubbles and in the form of iso-value maps. These results are then discussed by comparison with finite element calculations that are described later. It should be noted that, although finite element calculations generally give accurate results for linear problems, they may nevertheless sometimes depart from the exact solution.

In particular, finite element solutions around voids are very mesh sensitive. Numerical results need therefore be considered carefully when compared with the analytical derivations.

The equivalent inclusion method of Eshelby

Eshelby has established the following fundamental result. Let D be an elastic and isotropic infinite body. Consider Ω as a part of D, small in comparison with D. We can isolate fictitiously Ω which is supposed to represent an ellipsoidal inclusion. Now remove this inclusion from D and impose to it an eigenstrain β , that has no relation with any stress: for example, a thermal deformation or a phase transformation. If the inclusion is replaced into D, which has not been transformed yet, its deformation is no longer free due to the influence of the surrounding matrix. Thus, a stress field appears in Ω and D. The inclusion deformation, ξ , due to the presence of the matrix D, is related to the eigenstrain by β = ξ : S , where S is the fourth order Eshelby tensor. Through this derivation, Eshelby showed that the deformation of an ellipsoidal inclusion embedded in an infinite homogeneous medium, submitted to uniform remote loading, is homogeneous. This result allows the inhomogeneity problem to be dealt with. Consider now an ellipsoidal region, referred to as an inhomogeneity, in an infinite medium, with elastic constants different from the rest of the material.

The Eshelby result is applied to the inhomogeneity. Consider the infinite elastic body of Hooke tensor C , submitted to 0 σ and the corresponding strain 0 ε at infinity. The ellipsoidal inhomogeneity Ω with Hooke tensor i C disturbs locally the stress field. The aim of the analysis is to determine the perturbations caused by this inhomogeneity. The basic idea of Eshelby is to substitute to the inhomogeneity an homogeneous inclusion with the same properties as the matrix, but submitted to an eigenstrain. The eigenstrain must be determined such as to produce the same stresses and strains as the former inhomogeneity. In the inhomogeneity, the elastic strain is ξ + ε 0 , whereas in the equivalent inclusion it is given by β -ξ + ε 0 . The equivalence condition for the stresses in the inhomogeneity and the inclusion is therefore:

( ) ( ) β - ξ + ε = ξ + ε 0 0 i : C : C (1)
which, combined with β = ξ : S allows the eigenstrain tensor β to be determined.

Extension of the equivalent inclusion method of Eshelby

While the problem of interacting inhomogeneities cannot be solved in closed form (the "exact" solution involves infinite series expansions [START_REF] Mura | Micromechanics of defects in solids[END_REF]), an approximate analytical derivation is proposed below. In a first step, the interaction of two bubbles with equal internal pressures will be dealt with. Using the superposition principle illustrated in Fig. 1 (a), (b), and (c), it is sufficient to consider merely cavities. Application of the equivalent inclusion method extended to two inclusions then leads accordingly to introduce two eigenstrains, β I and β II , that depend on the space coordinates. For the sake of simplicity, they will be approximated by their values at the centers of the respective bubbles. In a second step, the effect of a pressure difference ∆p between the two bubbles will be addressed in the form of a small perturbation to the above solution. It is worth to note that the boundary conditions at the bubble surfaces, i.e. σ rr =p and σ rθ = 0, are not strictly fulfilled by such an estimation of the exact solution.

Solid inhomogeneities will first be considered, but in further derivations their elastic constants will be set equal to zero in order to deal with cavities. The problem, which is axisymmetric, will be solved in a plane containing the centre of the two inhomogeneities. In

II I 0 I η + ξ + ε = ε (2)
with

I I : S β = ξ ( 3
)
where S is constant and the same for the two inclusions since its expression depends only on the shape of the inclusions, which are both spherical here, and

II II II : D β = η (4)
where II D is the influence tensor associated with inclusion II. Thus, the strain II η represents the influence of inclusion II on inclusion I. In the same way, the influence of inclusion I on inclusion II is I η .

Note that the eigenstrains could be expanded in polynomial series, e.g.

...

x x B x B B ) ( l k II ijkl k II ijk II ij II ij + + + = β x (5)
As β = ξ : S and β = η : D , they could be expanded in polynomial series in the same way.

However, to a first approximation, the eigenstrains β I and β II will be assumed uniform, whence:

I I : S β = ξ (6)
is constant while

II kl II ijkl II ij B ) ( D ) ( x x = η (7)
remains a function of the space variables through the influence tensor.

The equivalence conditions for the stresses in the two inhomogeneities and the two inclusions can now be written:

( ) ( ) ( ) ( )      β - η + ξ + ε = η + ξ + ε β - η + ξ + ε = η + ξ + ε II I II 0 I II 0 II I II I 0 II I 0 I : C : C : C : C in Ω 1 in Ω 2 (8)
Since the material is isotropic, the constitutive equation can be written in the following form:

( ) ij kk ij ij 3 µ 2 µ 2 δ ε - κ + ε = σ (9)
where κ and µ are the bulk and shear moduli, respectively.

Equation ( 8) then becomes:

( ) ( )( ) ( ) ( ) ( ) ij
ij I D S 3 µ 2 D S µ 2 D S 3 µ 2 D S µ 2 δ β - β + β + ε - κ + β - β + β + ε = δ β + β + ε - κ + β + β + ε in Ω 1 (10) ( ) ( ) ( ) ( ) ( ) (
ε + β + β + κ - ε + β + β δ = ε + β + β -β + κ - ε + β + β -β δ in Ω 2
As seen above, the Eshelby tensors are the same for Ω 1 and Ω 2 . The above system must now be solved in order to determine the eigenstrains for each inclusion.

Analytical resolution

Derivation of the influence tensors D ijkl

The expression of D ijkl for a spherical inclusion of radius a is given by Mura [START_REF] Mura | Micromechanics of defects in solids[END_REF]:

[ ] jk li ik lj jl ki il kl ij kl klij ijkl , , , , ) 1 ( , 2 , ) ( D ) 1 ( 8 δ
Φ + δ Φ + δ Φ + δ Φ ν - - Φ νδ - Ψ = ν - π x (11)
with [ ]

) ( I x x ) ( I 2 1 K k k λ - λ = Φ and
( ) Here, the following convention is used: summation from 1 to 3 is extended over repeated lower case indices; capital indices take the same values as the corresponding lower case ones but without summation. Note that, in equation ( 13), the number of indices n=0, 1 or 2.

2 i i k k K I k k IK x , I( ) x x I ( ) a I ( ) x x I ( ) 2   Ψ = λ - λ - λ - λ   (12)
As seen before, the unknown eigenstrain tensors β . This system will be solved for two inclusions centered at (0,0,0) and (0,0,d). The (constant) values of the two eigenstrain tensors are those derived at the center of the associated inclusion, although such a choice is somewhat arbitrary since they could be calculated anywhere in the inclusions. The derivation is detailed below for one inclusion, Ω 1 . Thus, owing to axisymmetry, η can be written in the following form: and all the other S ijkl =0.

          η η η η η η η = η
D 1111 = D 2222 = ) 1 ( 30 ) 2 1 ( 5 9 3 2 5 2 ν - ν - α + α D 3333 = ) 1 ( 15 ) 2 ( 10 15 3 2 5 2 ν - ν - α - α D 1122 = D 2211 = ) 1 ( 30 ) 2 1 ( 5 3 3 2 5 2 ν - ν - α - α D 3311 = D 3322 = ) 1 ( 15 ) 2 1 ( 5 6
The system [START_REF] Neuber | Kerbspannungslehre[END_REF] can now be solved. For i and j=1,3, it leads very easily to:

II 13 I 13 β = β =0 (16)
so that the system reduces to four equations for only four unknowns. The elastic moduli µ I , µ II , κ I and κ II are now set equal to zero, since cavities are considered, and the hydrostatic loading is accounted for by specifying 0 ij ε =(p/3κ)δ ij . The following system is obtained: The strain field in the matrix associated with the case of Fig. 1 (b) can now been written in the form:

       - = β + β + β + β - = β + β + β + β - = β + β + β + β - = β + β + β + β p A A A A p A A A A p A A A A p A A A
II 3 2 1 II I 3 2 1 I 0 : ) x , x , x ( D : ) x , x , x ( D β + β + ε = ε (18)
Owing to the axisymmetric configuration, the interactions are analyzed in the plane Therefore, only 12 components of the influence tensor need be calculated. The stresses for the case of two cavities with the same pressure p are then obtained from the constitutive equation [START_REF] Papkovitch | Solution générale des équations différentielles fondamentales d'élasticité, exprimées par trois fonctions harmoniques[END_REF]. Furthermore, the effect of a pressure difference between the two bubbles may be estimated to a first approximation by adding the stresses determined in (25) below.

It is worth to note that the general solution of the problem proposed here is only an estimation. Indeed we first assumed the eigenstrains uniform. The second approximation is related to the boundary conditions. In order to apply the principle of linear superposition, the tangential components due to the pressure difference between the bubbles are neglected.

Under such approximation, the equality between the boundary conditions illustrated in Fig. 1 remains correct.

For two cavities with the same internal pressure p, the solution of the case depicted in Fig. 1 (c) is:

0 c ij ε =(-p/3κ)δ ij (21) Since 0 c ij ε + 0 ij ε =0
, the strain field for two interacting cavities with the same pressure is the linear superposition of solutions for the problems illustrated in Fig. 1 (b) and 1 (c):

II 3 2 1 II I 3 2 1 I : ) x , x , x ( D : ) x , x , x ( D β + β = ε (22)
The case depicted in Fig. 1 (d) is easy to analyse since the stress field of an isolated bubble in an infinite elastic solid has an analytical expression. If an internal pressure ∆p is applied in the absence of remote loading, the displacement field exhibits spherical symmetry: u = u r (r) e r (23) where e r denotes the radial unit vector.

Resolution of the equilibrium equations then yields: ( )

u(r) =
33 22 11 m 3 1 σ + σ + σ = σ (26) ( ) ( ) ( ) [ ] ( ) 2
1 σ + σ + σ + σ - σ + σ - σ + σ - σ = σ (27)
Hence the distribution of the two above quantities for two cavities of different sizes with different pressures can be analyzed.

For the spherical coordinates, we change 1→ r, 2→ θ, and 3→ ϕ.

Results and discussion

It is important to note that for an isolated bubble, the mean stress vanishes at any point:

0 r a 2 p , r a p m 3 3 3 3 rr = σ ⇒ ∆ = σ = σ ∆ - = σ ϕϕ θθ (28)
It is therefore not possible to determine the influence of ∆p on the mean stress around the two bubbles because the contribution of the configuration illustrated in Fig. 1 (d) is zero. The above approach is expected, however, to give a good estimation of the von Mises equivalent stress.

The present method allows various configurations to be easily investigated. Three parameters can be changed: the size and pressure of each bubble, and the distance between the bubble centers. In the following, three different cases are considered. The first case (i) consists of two bubbles of same size with same internal pressure, the second one (ii) of two different bubbles of different sizes with same pressure, and the third one (iii) of two bubbles of same size with different pressures. In the two first cases the mean stress and the von Mises equivalent stress are analyzed, while for the third case only the von Mises equivalent stress can be discussed (see above). Two representations of the results are used for each case: the stresses are displayed both along the symmetry axis and on a map in the half plane (x 1 ,x 3 ). To discuss the analytical approach, the results are compared with numerical calculations carried out with the Abaqus® software. An example of the mesh used for two bubbles of equal size is given in Fig. 3. It is divided into two different regions: in the first one, close to the cavities, where the stress fields are to be determined precisely, the elements are small. The second one, far away from the cavities, is made of larger elements. To obtain the best results, the mesh must be very fine and the elements must be close to squares. Moreover, for ensuring a good representation of an infinite solid, the mesh is very large compared to the bubble sizes and infinite elements are used far away from the bubbles. These requirements are fulfilled in the mesh. In Fig. 4, the stress distributions along the symmetry axis are displayed for the two first cases. The corresponding maps are given in Fig. 5 and6. The last case is shown in Fig. 7. All calculations were carried out using the elastic constants of palladium, which is usually employed to store nuclear fuels, i.e. κ=171.3 GPa, µ=42.7 GPa, and ν=0.385.

As shown on the curves and the maps, there are slight differences between the analytical and numerical values of the mean stress, which are very small with respect to the von Mises equivalent stress. This is because the nonzero mean stress is induced by the interaction of the bubbles (it vanishes for an isolated bubble), whereas this interaction only introduces a perturbation of the von Mises equivalent stress.

The errors brought to the exact solution by the above-mentioned approximations can be estimated by considering the values of the normal stresses at the bubble surfaces, in particular along the symmetry axis x 3 . The discrepancies between the exact values (i.e., rr p σ = -) and that predicted by the analytical approach are less than 1% for the three investigated configurations. The largest errors (overestimations) occur of course at points where the two bubbles are facing. Moreover, the gap is significantly smaller along the equators of the bubbles.

In Fig. 4, it is apparent that the analytical curves are very close to the numerical results except for a few points where the latter are slightly larger than the analytical predictions. The assumption of uniform eigenstrains is likely to be responsible for this discrepancy. Similarly, the von Mises equivalent stresses derived analytically (see Fig. 5 and6), exhibit local maxima along the x 1 and x 3 axes, that can be related to the Taylor first order expansions. This first analysis therefore shows that, despite its simplifying assumptions, the present approach leads to analytical results which are quite similar to the finite element predictions.

The mean stress maps allow the material regions undergoing tensile or compressive loading to be easily localized. Such results can be taken into account in a diffusion model. Helium atoms, after being created by tritium decay, diffuse in the material towards the bubbles. From the point of view of mechanics, the evolution of a pair of bubbles of different sizes can be predicted from the mean stress map in Fig. 6. The tension mean stress is larger near the smaller bubble than near the larger one. Helium atoms will therefore diffuse preferentially towards the small bubble, until the latter reaches the same size as the larger one.

It is thus likely that the bubbles will tend to progressively equalize their sizes. This is a part of the mechanism of diffusion, which should be added to a thermodynamical analysis to obtain a complete description of the evolution of the system.

The above results can also be used to predict the occurrence of cracks in the vicinity of the bubbles. The levels of the von Mises equivalent stresses also give quite an important information about the areas where plasticity occurs, i.e. at each point where the von Mises equivalent stress is larger than the yield stress (0.23 GPa) of the material. Therefore, a complete mechanical description of the material evolution will require to take plasticity into account.

Conclusions

A new approach for the estimation of the elastic fields between two bubbles loaded by an internal pressure was proposed. In spite of its simplicity, it leads to analytical results, which are in good agreement with finite element calculations. The diameters and pressures of the two bubbles can be easily varied, such that various configurations can be investigated straightforwardly. The two main results are the following:

(i) According to the mean stress distributions, the mechanical contribution to the diffusion process is likely to progressively equalize the sizes of the neighbouring bubbles.

(ii) The levels of the von Mises equivalent stresses indicate the areas where plasticity must be taken into account for a complete mechanical description of the system.

= + τ - + τ α α = - τ + + τ α α - = + τ - + τ α α = - τ + + τ α α - = - τ = + τ = ( ) ( ) ( )( ) [ ] { } ( )( ) [ ] { } 32
+ τ + τ + τ + α τ + τ + τ + α α + τ + τ + τ - α α + τ + τ + τ + α α τ + τ + τ - α α + τ + τ + τ + α α + τ + τ + τ - α α τ + τ + τ = ( ) ( ) ( ) ( ) ( ) ( ) 64
+ τ + τ + α α + τ + τ - α α + τ + τ + α α + τ + τ + α α + τ + τ - α α τ = Similarly, = β II
+ τ + τ + τ + α τ + τ + τ - α α + τ + τ + τ - α α + τ + τ + α α τ + τ - α α + τ + τ + τ + α α + τ + τ + τ - α α τ + τ + τ = ( ) ( ) ( ) ( ) ( ) ( ) 64
-analytical method, + numerical method σ M i se s (GPa)

x 3 (nm)

-analytical method, + numerical method von Mises equivlent stress map (GPa)

Analytical Numerical

Fig. 7

Fig. 1 , 1 greater

 11 Fig.1, the transition from gas bubbles to cavities without internal pressure is shown, with p 1

β

  are assumed to be uniform in each inclusion. Owing to axisymmetry, the expressions of the eigenstrains can be simplified (Fig. 2): Since β 11 =β 22 , β 13 =β 23 and β 12 =0, only three eigenstrain components are to be determined, namely β 11 , β 33 and β 13. The linear system (10) then decomposes into one set of four equations for the "diagonal" unknowns I 11 β , I 33 β , II 11 β , and II 33 β on the one hand, and a set of two equations for the "non-diagonal" unknowns I 13 β and II 13

  only 3x6=18 D ijkl components are needed:

  all the other D ijkl =0.In the above equations, α 2 =a 2 /d, and for the second inclusion the corresponding components are the same with α 1 =a 1 /d.4.2. Derivation of the eigenstrains and the stress fieldAs seen above, ξ can be written in the following form:

  A ij , given in the Appendix, depend on the components of the Eshelby and influence tensors. Once the eigenstrains are determined (see Appendix), the interaction fields between the inclusions can be calculated with the help of the influence tensors

(x 1 ,x 3 )

 13 . Since the eigenstrain tensor has the following diagonal form: components of the influence tensor are D ij11 and D ij33 . The interaction strains for the case of Fig.1(b) are given by:

  stress and the von Mises equivalent stress are derived:

  expressions of the eigenstrains β ij .

  of α 1 and α 2 .

Figure captions Fig. 1 :

 captions1 Figure captions

Fig. 2 :

 2 Fig. 2: The two cavities and the set of Cartesian coordinates.

Fig. 3 :

 3 Fig. 3: The finite element mesh: (a) General view; (b) the area close to the cavities.

Fig. 4 :

 4 Fig. 4: (a) Mean stress along axis x 3 for two bubbles of same size with same pressure; (b) Mean stress along axis x 3 for two bubbles of different sizes with same pressure; (c) von Mises equivalent stress along axis x 3 for two bubbles of same size with same pressure, and (d) von Mises equivalent stress along axis x 3 for two bubbles of different sizes with same pressure.

Fig. 5 :Fig. 6 :Fig. 7 :

 567 Fig. 5: Mean stress and von Mises equivalent stress maps for two bubbles of same size with same pressure

  Fig.3

Fig. 4 x

 4 Fig.4

  )

	2µ	II	0 ij	S	II ijkl kl	D	I ijkl kl I	II	II 2µ 3	0 kk	S	II kkmn mn	D	I kkmn mn I	ij
	2µ	0 ij	S	II ijkl kl	D	I ijkl kl I	II ij	2µ 3		0 kk	S	II kkmn mn	D	I kkmn mn I	II kk	ij

APPENDIX

a) Analytical expressions of the A ij coefficients of the linear system (17).

( ) (

)

) ( )

Analytical Numerical

von Mises equivalent stress (GPa), a=0.5nm, d=2.5nm (a/d=0.5), p 1 =p 2 =0.5GPa

Analytical Numerical

Fig. 5 mean stress map (MPa), a 1 =0.25nm, a 2 =0.75nm, d=2.5nm, p 1 =p 2 =0.5GPa

Analytical Numerical

von Mises equivalent stress (GPa), a 1 =0.25nm, a 2 =0.75nm, d=2.5nm, p 1 =p 2 =0.5GPa

Analytical Numerical

Fig. 6