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# Development of a robust methodology for the design assessment of bonded reinforcements on steel structures

Quentin Sourisseau

► To cite this version:

Quentin Sourisseau. Development of a robust methodology for the design assessment of bonded reinforcements on steel structures: Use of cohesive zone model and distributed optical fiber for characterization. Engineering Sciences [physics]. Nantes Université, 2022. English. NNT: tel-04282217

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https://hal.science/tel-04282217

Submitted on 16 Nov 2023

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# THESE DE DOCTORAT DE

## NANTES UNIVERSITE

ECOLE DOCTORALE N° 602

Sciences pour l'Ingénieur

Spécialité : Mécanique des Solides, des Matériaux, des structures et des surfaces.

Par

Quentin SOURISSEAU

## Development of a robust methodology for the design assessment of bonded reinforcements on steel structures

« Use of cohesive zone model and distributed optical fiber for characterization »

Thèse présentée et soutenue à Nantes, le 21/10/2022

Unité de recherche : Département Matériaux et Structures (MAST), Université Gustave Eiffel

## Rapporteurs avant soutenance :

Evelyne Toussaint

Professeur, Polytech Clermont Ferrand, Université Clermont-Auvergne

Emmanuel Ferrier

Professeur, Université Lyon I

## Composition du Jury :

Président :

Pascal Casari

Professeur, IUT St Nazaire, Nantes Université

Jury :

Evelyne Toussaint

Professeur, Polytech Clermont Ferrand, Université
Clermont-Auvergne

Emmanuel Ferrier

Professeur, Université Lyon I

De Barros Silvio

Professeur, Ecole Technique de Rio de Janeiro

Peter Davies

Ingénieur HDR, Ifremer

Dir. de thèse :

Sylvain Chataigner

Ingénieur-chercheur IPEF-HDR, Université Gustave Eiffel

Encadrants :

Emilie Lepetre

Ingénieur-chercheur ITPE, Université Gustave Eiffel

Xavier Chapeleau

Chargé de recherche, Université Gustave Eiffel - I4S
team Inria
**Titre : Développement d'une méthodologie robuste pour l'évaluation et la conception des renforts collés sur acier**

« Utilisation des modèles de zone cohésive et de mesures réparties de déformation par fibre optique pour la caractérisation »

**Mots clés** : collage, éléments cohésifs, méthode aux éléments finis, capteurs continus à fibre optique, méthode de la J-intégrale, corrélation d'images numériques.

**Résumé :**

Ce travail de thèse porte sur le développement et l'évaluation de méthodes de dimensionnement adaptées aux renforts collés sur structures métalliques en milieu offshore en se concentrant sur des assemblages métal-métal. Les méthodes proposées reposent sur l'utilisation de mesures par fibre optique continue pour l'étape de caractérisation basée sur des investigations de mécanique de la rupture, et sur les modèles de zone cohésive pour la modélisation de l'assemblage collé. Elles ont été éprouvées à travers plusieurs investigations expérimentales en mode I, mode II et mode mixte, et des études numériques via l'utilisation d'éléments finis dédiés. L'ensemble des méthodologies a ensuite été appliqué sur des assemblages collés de taille réelle afin de vérifier leur capacité prédictive. Les résultats des investigations expérimentales ont été comparés aux prédictions numériques des méthodes développées ainsi qu'à une autre approche reposant sur l'utilisation de critères couplés contrainte-énergie.

**Title: Development of a robust methodology for the design assessment of bonded reinforcements on steel structures**

« Use of cohesive zone model and distributed optical fiber for characterization »

**Keywords:** adhesive bonding, cohesive zone modeling, finite element method, distributed optical fiber sensor, J-integral method, digital image correlation.

**Abstract:**

This thesis work focuses on the development and the assessment of design methods adapted to reinforcements bonded on steel structure in offshore environment, and more specifically to steel-to-steel bonded joints. The proposed methodologies rely on the use of continuous optical fiber measurement during the characterization step based on fracture mechanics investigations, and the use of cohesive zone modeling for the bonded joint modeling. They were proven through several experimental investigations in mode I, mode II and mixed mode, and through numerical investigations using finite elements. The developed methodologies were then applied to real-scale samples in order to verify their predictive capacities. Experimental results were compared to numerical predictions and to an alternative approach from the literature relying on the use of coupled stress-energy criteria.
Remerciements

Il existe plusieurs citations qui mériteraient de trouver une place dans le préambule de ce
paragraphe pour résumé l'aboutissement de ce travail de thèse. Cependant la conclusion de
monsieur De la Fontaine retraçant l'histoire d'un charretier mal en point semble très à propos :
« aide-toi et le ciel t'aidera ». Cette phrase est en effet la directe expression du difficile contexte
dans lequel ces trois ans de travaux ont été effectué. À savoir, que s'il fallait attendre les actes
d'un grand barbu pour avancer, on n'aurait pas été bien loin...

Le document qui suit revêt la signature d'une seule personne, mais c'est le résultat de travaux
effectués par plusieurs qu'il est nécessaire de faire apparaitre dans ces quelques lignes pour
rendre honneur aux qualités humaines, techniques et scientifiques qu'ils ont déployés pour
permettre l'aboutissement de ces recherches, et la rédaction de ce manuscrit.

Pour ne pas déroger à l'ancestrale coutume dictant l'ordre d'apparition de ces personnes,
j'adresse un sincère remerciement à Sylvain Chataigner, Emilie Leprêtre, Xavier Chapeleau,
ainsi qu'à Stéphane Paboeuf et Luc Mouton. Le fait que ces lignes existent, prouvent que vous
avez su canaliser le turbulent et pas toujours très rigoureux doctorant des premiers mois, et que
vous m'avez donné l'opportunité de travailler et apprendre à mon rythme. Merci de la confiance
que vous m'avez accordée pour mener ce doctorat au bout et de la qualité du cadre de travail
que vous m'avez fourni malgré les innombrables difficultés contextuelles.

Je remercie également les membres du jury, le président Pascal Casari, les rapporteurs Evelyne
Toussaint et Emmanuel Ferrier et enfin les examinateurs Silvio de Barros et Peter Davies, ainsi
que le membre de mon CSI Romain Créac'hcadec. Je leur suis reconnaissant du temps qu'ils
ont bien voulu consacrer à l'analyse de mes travaux.

Ce travail a été réalisé conjointement entre le laboratoire Structures Métalliques et à Câbles (SMC) de l'Université Gustave Eiffel et l'équipe Matériaux Composites du département de recherche de Bureau Veritas, j'adresse mes remerciements à ces deux équipes pour leurs accueils et moments passés ensemble. Plus précisément, je remercie, Jean-François David, Richard Michel, Pauline Bochereau, Yannick Falaise et Ludwig Battais pour leurs disponibilités permanentes et aide à la réalisation de la partie expérimentale de ce doctorat. Je remercie Benjamin Collier et Jean-Christophe Petiteau sur la transmission de leur savoir technique et théorique lié aux problématiques du collage. Je remercie aussi Jean-Philippe Terrier et Vincent Gaudrefroy du laboratoire MIT pour leur aide lors de la réalisation de l'étude sur les propriétés des préparations de surface de la campagne expérimentale de la thèse. Et finalement, pour clore la partie professionnelle de ce paragraphe, un remerciement particulièrement appuyé à Maxime Deydier pour la qualité et l'efficacité de son travail produit au cours de ce doctorat, sans qui et sans quoi, les résultats et analyses présentés dans ce document n'auraient pas pu être aussi approfondis.

Je remercie aussi ma famille, les frangins/frangines, Fabien et Manon, tonton Michel, Mamie et plus particulièrement mes parents, Raphaëlle et Jean-François, qui m'ont toujours soutenu et sans qui, évidemment, je ne serais pas celui que je suis aujourd'hui. Merci maman pour ta permanente gentillesse et abnégation dont seule une mère est capable et qui a permis mon développement personnel. Merci papa, scientifique jusqu'au bout des ongles pour tes conseils toujours avisés et soutien permanent quand il y en a eu besoin.

Finalement, un pensé aux copains qui si un jour, lisent ces lignes, se reconnaitrons. Et plus particulièrement Gabrielle pour m'avoir accompagné pendant cette dernière année de thèse et qui a contribué, à sa façon, à la réussite de ce doctorat.
# Résumé en français

Cette thèse s'inscrit dans le cadre du projet Strength Bond Offshore dédié au développement d'une réparation de tôles d'acier corrodées à l'aide d'un patch composite pour augmenter la durée de vie de structures offshore. Le développement d'une réparation composite pour ce type de structures semble particulièrement approprié pour diverses raisons. Premièrement, ces matériaux présentent des performances élevées, peuvent être facilement adaptés à diverses géométries tridimensionnelles, et sont faciles à transporter et à installer. Ils sont en outre non corrodables et donc particulièrement bien adaptés au milieu offshore. Enfin, cette technique ne nécessite aucun travail dit "à chaud" tel que la technique de réparation commune : le découpage (généralement au plasma) des tôles corrodées et le soudage de nouvelles tôles. Ceci est particulièrement important dans le cas de la réparation ou du renforcement des FPSO (Floating Production, Storage and Offloading).

Cependant, l'absence d'approches de dimensionnement communément adoptées et validées pour ce type de renforcement conduit à un surdimensionnement des joints collés dans les applications industrielles diminuant l'intérêt de la méthode, voire limitant le recours à cette technique dans le cas de géométries complexes. Une meilleure compréhension des techniques de dimensionnement est donc nécessaire ainsi qu'une évaluation de leur robustesse. Une méthodologie de caractérisation et de conception des joints collés est donc proposée dans cette étude. Elle a été validée sur le cas d'assemblages collés métal-métal représentatifs d'applications industriels. Elle pourrait servir par la suite de base pour développer une méthodologie adaptée au cas des renforts composites collés.

La première partie du document s'attache à la réalisation d'un état de l'art sur les joints collés structuraux. Des éléments généraux sur le sujet du collage structural sont tout d'abord rappelés. On retiendra en particulier la notion d'adhésion dépendante des opérations de préparation de surface et impactant les modes de rupture obtenus.

Une revue des différents modèles mécaniques existants pour représenter le comportement d'assemblage collé est ensuite réalisée. Les principaux modèles analytiques sont ainsi rappelés : les modèles de Volkersen et de Goland et Reissner dans le cas d'un adhésif ayant un comportement élastique, et de Hart-Smith dans le cas d'un adhésif élasto-plastique. Ces modèles permettent d'obtenir de façon directe l'état de contraintes au bord d'un joint collé dans le cadre de géométries simples. Cependant, avec des assemblages collés de géométries plus complexes ou constitués de matériaux ayant un comportement mécanique fortement non linéaire, il est souvent nécessaire de recourir à l'utilisation de méthodes numériques. Plusieurs approches sont également présentées dont celle des éléments finis qui sera utilisée dans la suite de l'étude. On retiendra que dans ce cas, l'étude des assemblages collés présente une difficulté liée à la présence de singularités physiques responsables de problèmes de convergence des modèles.

Ensuite, une revue des critères de rupture utilisés pour le dimensionnement des assemblages collés est réalisée. Trois familles principales de critères sont identifiées : les critères en contrainte ou déformation relevant de la mécanique des milieux continus, les critères en énergie relevant de la mécanique de la rupture, et les approches couplées. Ces dernières présentent l'avantage de pouvoir s'intéresser à la fois à des structures continues et fissurées. Pour cette raison, elles semblent être un bon compromis pour le dimensionnement des joints collés. Elles seront ainsi utilisées dans le cadre de notre étude. Deux méthodes principales sont exposées : la méthode couplée contrainte-énergie et la méthode des modèles de zone cohésive qui implique
l'identification des lois d'interface. C'est cette dernière méthode qui sera plus particulièrement utilisée dans notre travail. Elle repose sur des investigations expérimentales visant à déterminer la ténacité ou énergie de déformation à rupture, et sur l'utilisation d'outils de modélisation aux éléments finis cohésifs.

Le second chapitre de la thèse porte sur l'amélioration des investigations expérimentales visant à caractériser la ténacité des assemblages collés via des mesures réparties de déformation par fibre optique. Celles-ci doivent permettre de suivre la propagation d'une fissure dans un assemblage collés en étant complètement indépendante de l'opérateur. Cette mesure est effectuée de manière continue pendant l'essai, et permet de connaître la longueur de fissure au milieu du joint collé. Plusieurs stratégies d'instrumentation sont présentées et évaluées via des investigations expérimentales, des modèles analytiques et des modèles numériques. Cette étude est réalisée pour les deux modes principalement rencontrés dans les applications offshore (mode I et mode II) et pour le mode mixte. L'essai DCB (Double Cantilever Beam) a été utilisé pour le mode I. L'essai ENF (End Notched Flexure Test) a été utilisé pour le mode II. L'essai MMB (Mixed Mode Bending) avec deux ratios de mode différents a été utilisé pour le mode mixte. Pour ces trois essais, les géométries d'éprouvettes et de bâtis d'essai sont inspirées des normes existantes pour ces différents tests et adaptées au cas étudié. Le suivi du front de fissure par fibre optique a été validée avec des éprouvettes métal-métal collés avec une résine époxy (désignée par la référence A pour une raison de confidentialité vis-à-vis du projet Strengthbond).

L'utilisation d'une fibre optique continue collée en surface des éprouvettes et le développement d'un post-traitement adapté des mesures de déformation a permis de réaliser pour ces trois essais un suivi de la propagation de fissure. Les résultats ont été comparés à des modèle numériques et des mesures visuelles après rupture et une bonne corrélation a été constatée. L'intérêt de la méthode au regard de la forme du front de fissure a également été montrée. Une limitation existe cependant dans le cas d'une propagation de fissure trop rapide (ou incontrôlée) avec les moyens de mesure à notre disposition. L'utilisation d'une fibre optique continue insérée au sein de la couche de colle a également été étudiée. Les résultats ont été moins satisfaisants et il semble que davantage d'investigations soient nécessaires pour mieux relier les profils de déformation aux phénomènes physiques se produisant dans la colle lors de la propagation d'une fissure. Cette étude a également mis en avant que les capteurs à fibre optiques continus peuvent être utilisés pour d'autres type de chargements, ou de géométrie d'éprouvettes.

Le troisième chapitre de la thèse porte sur la détermination des ténacités d'un assemblage collé métal-métal lors d'essais de mécanique de la rupture (essais DCB, ENF et MMB) par des méthodes normalisées ou en utilisant la méthode précédemment développée pour le suivi de propagation de fissure à l'aide de fibre optique continue. Cette étude a été réalisée avec des éprouvettes métal-métal collés avec une résine époxy B qui a été utilisée pour toute la suite des travaux de thèse. En utilisant la fibre optique pour le suivi de fissure au cours des essais, les méthodes de détermination de la ténacité ont été adaptées à partir des normes et comparées à une autre méthode non normalisée : la méthode de l'intégrale J. Cette dernière permet d'obtenir l'état énergétique aux abords d'une pointe de fissure via des mesures de déplacements locaux (mesures réalisées à l'aide de la corrélation d'images numériques).

Les méthodes CCM (Compliance Calibration Method) et MBTM (Modifier Beam Theory Method) proposée par les normes se sont avérées efficaces pour obtenir la ténacité « à l'amorçage » de la fissure pour les essais DCB, ENF et MMB. Cependant, ces valeurs de ténacité dite d'« amorçage » sont connues comme étant faibles (ou conservatrices) par rapport à la ténacité dite de « propagation » qu'il est possible de mesurer avec l'instrumentation par fibre optique. Toutefois, du fait du choix de la résine pour cette étude, seuls les essais DCB ont
permis d'avoir une propagation de fissure contrôlée et donc de pouvoir appliquer les méthodes développées de mesure de ténacités à l'aide du suivi de fissure par fibre optique.

L'approche de l'intégrale J constitue également un moyen efficace d'obtenir la ténacité de propagation. Les résultats obtenus par cette méthode sont proches des résultats précédents obtenus par les méthodes CCM et MBTM modifiées dans le cas d'une instrumentation par fibre optique. La dérivation de la courbe de l'intégrale J a permis, en outre, d'évaluer directement les lois d'interface (modèles de zones cohésives) pour les modes I et II.

Le quatrième chapitre étudie deux approches de détermination des modèles de zone cohésive (pouvant être implémentées par éléments finis) ainsi que la démarche de modélisation associée. La première méthode est une calibration directe du modèle, basée sur l'analyse de l'intégrale J décrite précédemment. La deuxième méthode est une méthode indirecte basée sur la ténacité et s'appuyant sur l'utilisation itérative de modélisations aux éléments finis. La première partie de ce chapitre présente plus en détails la stratégie de modélisation par éléments finis et plus particulièrement la modélisation dans le cas de modes mixtes. Elle permet en particulier de mettre en exergue pour l'outil utilisé, les interdépendances des lois cohésives en mode I et en mode II et le rôle du coefficient de couplage $\alpha$. La deuxième partie décrit plus précisément les méthodes directes (lois cohésives obtenues avec l'intégrale J) et indirectes (par minimisation) utilisées dans ce travail pour déterminer les lois cohésives en mode I, II et en mode mixte. Ces deux méthodes sont ensuite appliquées aux résultats des trois séries d'essais, DCB, ENF et MMB, présentées dans le chapitre 3. Pour la forme de loi cohésive, la loi bilinéaire a été retenue dans cette étude car elle correspond assez bien à celle obtenue par la méthode directe. De plus, d'un point de vue numérique, elle présente l'avantage de ne nécessiter que deux paramètres à déterminer (la contrainte critique et la ténacité critique) dans le cas de la méthode indirecte, ce qui permet de limiter le nombre d'itérations par rapport à d'autres lois cohésives de formes plus complexes.

Néanmoins, les paramètres de la loi bilinéaire obtenus par la méthode indirecte sont assez différents de ceux obtenus par la méthode directe. Bien que les rigidités initiales soient similaires, les valeurs de la ténacité critique et la contrainte critique pour la méthode indirecte sont supérieures à celles obtenues par la méthode directe. Cela génère donc une différence dans la forme de la loi cohésive. D'un point de vue physique, la loi cohésive de la méthode directe semble plus réaliste car elle représente davantage le comportement fragile identifié pour l'adhésif étudié lors des essais. La forme de la méthode indirecte semble en contradiction avec cette observation puisqu'car elle présente une partie dédiée à l'endommagement plus importante et donc moins représentative d'un adhésif fragile. Le coefficient de couplage reste cependant proche entre les deux méthodes.

Les modèles cohésifs déterminés par les méthodes directes et indirectes ont ensuite été implémentées dans les modèles aux éléments finis pour simuler les essais de mécanique de la rupture réalisés (présentés au chapitre3). Les comparaisons entre modèles numériques et mesures expérimentales ont montré peu de différences entre les deux approches. Il semble donc que, dans le cas d'essais quasi-statique, la forme de la loi cohésive a peu ou pas d'impact sur le comportement et sur la prédiction de la charge à rupture des assemblages collés avec la résine B. Ceci est en accord avec plusieurs résultats de la littérature. Dans le cas d'une étude de la durabilité des assemblages collés, des études additionnelles seraient néanmoins nécessaires concernant le choix de la forme de ces modèles cohésifs.

Par ailleurs, il est important de souligner que la méthode indirecte nécessite un temps de calcul assez important par rapport à la méthode directe (via l'intégrale J). La méthode directe a permis de prédire plus précisément (erreur <10%) la charge à rupture des essais DCB, ENF et MMB. Elle implique néanmoins l'utilisation de davantage d'instrumentation, comme par exemple, dans notre cas, un système de corrélation d'images.
Suite à ces investigations portant sur les essais de mécanique de la rupture (pré-fissurés), le dernier chapitre s'est concentré à évaluer la capacité des méthodes développées pour prédire le comportement et la capacité d'assemblages collés plus réalistes (représentatif de solutions industrielles). Des essais sur assemblages à plus grande échelle ont donc été réalisés avec des conditions de chargement proches de la réalité (situations de chargement les plus couramment rencontrées pour les ponts de navires). Afin d'évaluer la pertinence de la méthodologie proposée et d'étudier sa robustesse, il a été décidé de comparer les résultats obtenus avec les méthodes directe et indirecte à ceux obtenus avec une méthodologie alternative s'appuyant à la fois sur la mécanique continue et la mécanique de la rupture : la méthode couplée contrainte-énergie.

Les essais ont été instrumentés par des jauges de déformation et une fibre optique continue afin de bien identifier l'amorçage du décollement et de suivre la propagation de la fissure. La fibre optique continue a effectivement pu suivre la propagation de la fissure. Ce résultat valide que ce moyen d'instrumentation peut aussi être appliqué en conditions réelles dans le cadre de monitoring ou de suivi qualité. Les résultats des essais se sont en outre avérés relativement répétables. Une capacité moyenne de l'assemblage collé pour les deux situations étudiées (traction et flexion) a ainsi pu être déterminée.

L'application de la méthode couplée contrainte-énergie a ensuite été réalisée. Son application, en particulier le critère en contrainte, s'est heurtée à plusieurs difficultés : le choix de la contrainte, la détermination du critère associé et le choix de la localisation. De fortes hypothèses ont été réalisées sur ce sujet, tendant à limiter l'applicabilité de la méthode pour des géométries plus complexes. Les deux méthodes indirectes et directes ont ensuite été appliquées à l'aide des résultats du chapitre 4.

Les résultats obtenus ont permis de valider la capacité des modèles cohésifs (avec la méthode directe et indirecte) et de la méthode couplée à prédire la charge à rupture de l'assemblage et son comportement local et général. La seule méthodologie dont les résultats sont conservateurs est la méthode directe. Le Tableau 0.1-1 compare les trois méthodes étudiées dans cette étude.

<table><caption>Tableau 0.1-1: Comparaison des méthodes de dimensionnement étudiée</caption><thead><tr><th></th><th>Tests requis</th><th>Temps de calcul (essais standard)</th><th>Temps de calcul (essais réels)</th><th>Données des modèles</th><th>Prédiction de la rupture</th></tr></thead><tbody><tr><th>Méthode directe</th><td>DCB/ENF/MMB<br>Avec instrumentation spéciale (DIC)</td><td>2 jours</td><td>4 jours</td><td>Comportement complet du patch et effort à rupture</td><td>Conservative</td></tr><tr><th>Méthode indirecte</th><td>DCB/ENF/MMB<br>Avec instrumentation standard</td><td>5-8 jours</td><td>4 jours</td><td>Comportement complet du patch et effort à rupture</td><td>Non conservative</td></tr><tr><th>Critère couplé</th><td>TAST/DCB/ENF/MMB</td><td>Ø</td><td>5-6 jours</td><td>Effort à rupture</td><td>Non conservative</td></tr></tbody></table>

Ce travail a permis d'établir et d'évaluer différentes méthodologies de dimensionnement des renforts collés sur acier pour des chargements typiquement rencontrés en conditions réelles. Ces méthodes se sont avérées prédictives pour le dimensionnement lors d'un chargement monotone, montrant que dans notre cas (avec la résine B), la forme des lois cohésives choisie avait peu d'impact sur cette prédictivité.

Les capacités prédictives des méthodes étudiées dans cette thèse doivent encore être évaluées vis-à-vis de la fatigue. De plus, dans le cadre du développement de renforts composites collés,
il sera nécessaire d'étendre une des méthodologies développées au cas d'interfaces multi-
matériaux. Des plis de fibre de verre sont en effet très souvent intégrés entre le renfort composite
à fibre de carbone et la structure métallique renforcée pour des raisons de corrosion galvanique.
Celui-ci peut avoir un impact non négligeable sur les propriétés d'interface.
# Details on StrengthBond Offshore JIP project

<figure><img src="image_2.png" /></figure>

Leading by **Stéphane Paboeuf** of Bureau Veritas (head of composite materials section research department).

Bureau Veritas has initiated the Strength Bond Joint Industrial Project (JIP), on the assessment of bonded repairs for offshore units. FPSO's have been around for the last forty years, with a large development in the early 1990's. About two hundred are now operated around the world. The offshore environment is extremely severe to these steel structures that are required to operate twenty years without dry docking. Corrosion is a permanent threat to FPSO's and maintenance is a challenge. The current crop and renewal repair techniques imply welding, thus:

- emptying, cleaning and inerting the surrounding tanks,

- opening a floating structure,

- rewelding the stiffeners, thus setting up large scaffoldings.

Mechanically, crop and renewal repairs imply large costs and down time. Composites and steel bonded repairs overcome these problems by being a non-intrusive and “cold” repair. It can be found surprising that such repairs have not been more developed in the industry. Offshore installation and durability are obvious problems to be treated. However, engineering face problems as basic as strength qualification. Analysis and design of a composite bonded repairs require tests that are delicate to translate to specific offshore design as there is a lack of guidance or reference. It makes difficult the qualification of a design of repair, and the evaluation of the safety margin.

Based on the experience gathered during the last twenty years, in research or certification of bonded repairs and composite projects, Strength Bond Offshore JIP has been initiated to overpass the limitation to qualification of bonded repairs. A clear objective of the JIP is to enable a first level of certification of bonded repairs. The workflow of the project is articulated between characterization and strength tests, to be compared to a large modelling work investigating the failure predictions techniques. Fatigue tests are to be performed as it is a necessary step to enable qualification of bonded repairs. Improved definition of failure criteria will allow the evaluation of the margin between the strength and the regulatory loads, hence definition of safety factors. Hence, bonded repairs design can follow classical offshore engineering routines. Approval of design will not require systematic tests and its duration will be shortened to allow acceptable time frames.

## Consortium of SBO JIP project :

<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /></figure>

Contact: **Stephane.paboeuf@bureauveritas.com**

Web page: [https://marine-offshore.bureauveritas.com/strength-bond-offshore-assessing-strength-bonded-repairs](https://marine-offshore.bureauveritas.com/strength-bond-offshore-assessing-strength-bonded-repairs)
# Contents

<table><tr><td>Remerciements</td><td>3</td></tr><tr><td>Résumé en français</td><td>4</td></tr><tr><td>Details on StrengthBond Offshore JIP project</td><td>9</td></tr><tr><td>Contents</td><td>10</td></tr><tr><td>Table of figures</td><td>15</td></tr><tr><td>Table of tables</td><td>21</td></tr><tr><td>General Introduction</td><td>24</td></tr><tr><td>Chapter 1: State of the art regarding the design of adhesively bonded joints</td><td>27</td></tr><tr><td>1.1. Generalities about adhesive bonding</td><td>27</td></tr><tr><td>1.1.1. Historical background</td><td>27</td></tr><tr><td>1.1.2. Adhesion theories</td><td>27</td></tr><tr><td>1.1.3. Failure modes</td><td>28</td></tr><tr><td>1.1.4. Load and geometries of adhesively bonded joint</td><td>29</td></tr><tr><td>1.2. Mechanical state analysis of bonded lap joints</td><td>30</td></tr><tr><td>1.2.1. Models based on analytical solutions</td><td>30</td></tr><tr><td>1.2.2. Models based on numerical approaches</td><td>40</td></tr><tr><td>1.3. Design approach and failure prediction</td><td>43</td></tr><tr><td>1.3.1. Continuum mechanics approach</td><td>43</td></tr><tr><td>1.3.2. Fracture mechanics</td><td>45</td></tr><tr><td>1.3.3. Coupled stress-energy method</td><td>49</td></tr><tr><td>1.4. Damage mechanics and cohesive zone modeling</td><td>53</td></tr><tr><td>1.4.1. Description and concept</td><td>53</td></tr><tr><td>1.4.2. Cohesive law shape</td><td>56</td></tr><tr><td>1.4.3. Determination of the critical energy release rate in mode I according to standardized tests</td><td>59</td></tr></table>
<table><tr><td>1.4.4.</td><td>Determination of the critical energy release rate in mode II according to standardized tests</td><td>62</td></tr><tr><td>1.4.5.</td><td>Determination of the critical energy release rate in mixed mode according to standardized tests</td><td>63</td></tr><tr><td>1.4.6.</td><td>Non-standard test Gc determination</td><td>65</td></tr><tr><td>1.5.</td><td>Conclusion</td><td>68</td></tr></table>

## 2. Chapter 2: A new methodology for fracture mechanics investigations on adhesively bonded joints relying on the use of distributed optical fiber sensor.................................. 71

2.1. Justification of the development of the new crack propagation measurement methodology

2.2. Optical fiber crack front measurement methodologies

2.2.1. Theoretical ENF crack front measurement

2.2.2. Theoretical DCB crack front measurement

2.2.3. Theoretical MMB crack front measurement

2.3. Description of the experimental protocol

2.3.1. Studied samples

2.3.2. DCB, ENF and MMB test bed geometries

2.3.1. Monitoring system by distributed optical fiber sensor

2.4. DCB test investigations

2.4.1. Load/displacement curve analysis and failure mode

2.4.2. Exterior optical fiber

2.4.3. Crack length visual validation

2.5. ENF test investigations

2.5.1. Load vs time curve analysis and failure mode

2.5.2. Exterior optical fiber analysis and crack propagation determination

2.5.3. Crack length visual validation

2.5.4. Embedded optical fiber curve analysis

2.5.5. Numerical validation of the crack length measurement

2.6. MMB test investigations

2.6.1. Load vs time curve analysis and failure mode
<table>
  <tbody>
    <tr>
      <td>2.6.2.</td>
      <td>Optical fiber analysis and crack propagation determination</td>
      <td>91</td>
    </tr>
    <tr>
      <td>2.6.3.</td>
      <td>Crack length visual validation</td>
      <td>92</td>
    </tr>
    <tr>
      <td>2.7.</td>
      <td>Conclusion</td>
      <td>92</td>
    </tr>
    <tr>
      <td colspan="2"><b>3. Chapter 3: Critical toughness and cohesive law experimental determination methodology</b></td>
      <td><b>95</b></td>
    </tr>
    <tr>
      <td>3.1.</td>
      <td>Realized experimental investigations</td>
      <td>95</td>
    </tr>
    <tr>
      <td>3.1.1.</td>
      <td>Samples and experimental program</td>
      <td>95</td>
    </tr>
    <tr>
      <td>3.1.2.</td>
      <td>Additional monitoring system: Digital Image Correlation</td>
      <td>97</td>
    </tr>
    <tr>
      <td>3.2.</td>
      <td>Determination of critical toughnesses using standard procedures</td>
      <td>98</td>
    </tr>
    <tr>
      <td>3.2.1.</td>
      <td>Obtained failure mode</td>
      <td>98</td>
    </tr>
    <tr>
      <td>3.2.2.</td>
      <td>Obtained results in mode I (DCB investigations)</td>
      <td>99</td>
    </tr>
    <tr>
      <td>3.2.3.</td>
      <td>Obtained results in mode II (ENF investigations)</td>
      <td>102</td>
    </tr>
    <tr>
      <td>3.2.4.</td>
      <td>Obtained results in mixed mode (MMB investigations)</td>
      <td>105</td>
    </tr>
    <tr>
      <td>3.3.</td>
      <td>Determination of critical toughnesses and cohesive laws using J integral for mode I and II</td>
      <td>106</td>
    </tr>
    <tr>
      <td>3.3.1.</td>
      <td>J-integral expression for DCB and ENF tests</td>
      <td>106</td>
    </tr>
    <tr>
      <td>3.3.2.</td>
      <td>Cohesive law expression for DCB and ENF tests</td>
      <td>110</td>
    </tr>
    <tr>
      <td>3.3.3.</td>
      <td>Analysis of the DIC strategy to monitor local displacement</td>
      <td>111</td>
    </tr>
    <tr>
      <td>3.3.4.</td>
      <td>Obtained results for DCB investigations</td>
      <td>115</td>
    </tr>
    <tr>
      <td>3.3.5.</td>
      <td>Obtained results for ENF investigations</td>
      <td>116</td>
    </tr>
    <tr>
      <td>3.4.</td>
      <td>Conclusion and discussion</td>
      <td>118</td>
    </tr>
    <tr>
      <td colspan="2"><b>4. Chapter 4: Description and evaluation of the adopted cohesive zone modeling strategy</b></td>
      <td><b>121</b></td>
    </tr>
    <tr>
      <td>4.1.</td>
      <td>Presentation of the used finite elements models</td>
      <td>121</td>
    </tr>
    <tr>
      <td>4.1.1.</td>
      <td>Cohesive elements</td>
      <td>121</td>
    </tr>
    <tr>
      <td>4.1.2.</td>
      <td>Finite element model hypothesis</td>
      <td>123</td>
    </tr>
    <tr>
      <td>4.1.3.</td>
      <td>Mode I and II cohesive laws determination methods</td>
      <td>125</td>
    </tr>
    <tr>
      <td>4.2.</td>
      <td>Cohesive zone model determination using indirect method</td>
      <td>127</td>
    </tr>
    <tr>
      <td>4.2.1.</td>
      <td>Calibration method</td>
      <td>127</td>
    </tr>
    <tr>
      <td>4.2.2.</td>
      <td>Cohesive laws reference parameters</td>
      <td>129</td>
    </tr>
  </tbody>
</table>
<table>
  <tbody>
    <tr>
      <td>4.2.3.</td>
      <td>Model results in mode II</td>
      <td>129</td>
    </tr>
    <tr>
      <td>4.2.4.</td>
      <td>Model results in mode I</td>
      <td>135</td>
    </tr>
    <tr>
      <td>4.2.5.</td>
      <td>Model results in mixed mode</td>
      <td>140</td>
    </tr>
    <tr>
      <td>4.3.</td>
      <td>Cohesive zone model determination using direct method</td>
      <td>142</td>
    </tr>
    <tr>
      <td>4.3.1.</td>
      <td>Model results in mode II</td>
      <td>143</td>
    </tr>
    <tr>
      <td>4.3.2.</td>
      <td>Model results in mode I</td>
      <td>144</td>
    </tr>
    <tr>
      <td>4.3.3.</td>
      <td>Model results in mixed mode</td>
      <td>145</td>
    </tr>
    <tr>
      <td>4.3.4.</td>
      <td>Resulting cohesive law parameters</td>
      <td>146</td>
    </tr>
    <tr>
      <td>4.4.</td>
      <td>Discussion</td>
      <td>146</td>
    </tr>
    <tr>
      <td><b>5.</b></td>
      <td><b>Chapter 5: Robustness appraisal of the proposed methodology</b></td>
      <td><b>149</b></td>
    </tr>
    <tr>
      <td>5.1.</td>
      <td>Experimental campaign at real scale and under most commonly encountered loading situations</td>
      <td>149</td>
    </tr>
    <tr>
      <td>5.1.1.</td>
      <td>Studied samples and instrumentation</td>
      <td>149</td>
    </tr>
    <tr>
      <td>5.1.2.</td>
      <td>Test protocols</td>
      <td>151</td>
    </tr>
    <tr>
      <td>5.1.3.</td>
      <td>Strain measurement and load at failure</td>
      <td>152</td>
    </tr>
    <tr>
      <td>5.2.</td>
      <td>Stress/energy criteria application to full size model</td>
      <td>155</td>
    </tr>
    <tr>
      <td>5.2.1.</td>
      <td>Hypothesis for FE model</td>
      <td>156</td>
    </tr>
    <tr>
      <td>5.2.2.</td>
      <td>Stress criteria and mesh sensitivity</td>
      <td>157</td>
    </tr>
    <tr>
      <td>5.2.3.</td>
      <td>Energy criteria</td>
      <td>160</td>
    </tr>
    <tr>
      <td>5.2.4.</td>
      <td>Load at failure prediction analysis using coupled stress-energy criteria</td>
      <td>163</td>
    </tr>
    <tr>
      <td>5.3.</td>
      <td>Application of the developed methodologies: direct and indirect method failure prediction</td>
      <td>165</td>
    </tr>
    <tr>
      <td>5.3.1.</td>
      <td>Hypothesis for FE model</td>
      <td>165</td>
    </tr>
    <tr>
      <td>5.3.2.</td>
      <td>Load/displacement curves</td>
      <td>166</td>
    </tr>
    <tr>
      <td>5.3.3.</td>
      <td>Strain measurements analysis during tension tests</td>
      <td>167</td>
    </tr>
    <tr>
      <td>5.3.4.</td>
      <td>Strain measurements analysis during bending tests</td>
      <td>171</td>
    </tr>
    <tr>
      <td>5.3.5.</td>
      <td>Crack front shape</td>
      <td>174</td>
    </tr>
    <tr>
      <td>5.4.</td>
      <td>Conclusion</td>
      <td>175</td>
    </tr>
    <tr>
      <td><b>6.</b></td>
      <td><b>General conclusion</b></td>
      <td><b>177</b></td>
    </tr>
  </tbody>
</table>
7. References.................................................................................................................. 180

8. Appendix.................................................................................................................. 190

A1: ENF specimens with primer – observations after failure.................................. 190

A2: TAST test of the resin B with primer................................................................ 191

A3: Surface preparation analysis ........................................................................ 192

A4: Tensile test result on resin B (bulk samples) .................................................. 198

A5: VCCT routine.................................................................................................... 199

A6: Full size tension and bending tests – observations after failure........................ 200
Table of figures

<table><tr><td>Figure 1.1-1: Failure modes of an adhesively bonded assembly.</td><td>28</td></tr><tr><td>Figure 1.1-2: Main types of loads applied on bonded joints.</td><td>29</td></tr><tr><td>Figure 1.1-3: Most commonly encountered bonded assemblies.</td><td>29</td></tr><tr><td>Figure 1.2-1: Constant shear model.</td><td>30</td></tr><tr><td>Figure 1.2-2: TAST Test [16].</td><td>31</td></tr><tr><td>Figure 1.2-3: Volkersen's model for the single lap joint.</td><td>31</td></tr><tr><td>Figure 1.2-4: Infinitesimal free-body-diagram of the Volkersen's model.</td><td>32</td></tr><tr><td>Figure 1.2-5: Shear stress along the bonded length according to Volkersen's theory for different values of ku/ka versus constant shear model.</td><td>33</td></tr><tr><td>Figure 1.2-6: Shear stress along the bonded length according to Volkersen's theory for different bonded length versus constant shear model.</td><td>34</td></tr><tr><td>Figure 1.2-7: Typical relation between ultimate capacity and overlap length [18].</td><td>34</td></tr><tr><td>Figure 1.2-8: Deformed state of a single lap joint in the case of the Goland and Reissner model [22].</td><td>35</td></tr><tr><td>Figure 1.2-9: Infinitesimal free-body-diagram of the Goland & Reissner model in the adherent out of the bonded area.</td><td>35</td></tr><tr><td>Figure 1.2-10: Infinitesimal free-body-diagram of the Goland & Reissner model in the joint.</td><td>35</td></tr><tr><td>Figure 1.2-11: Stresses predicted by Volkersen and Goland & Reissner models [22].</td><td>37</td></tr><tr><td>Figure 1.2-12: Application case of the Hart-Smith model.</td><td>38</td></tr><tr><td>Figure 1.2-13: Perfect plastic law of the adhesive.</td><td>39</td></tr><tr><td>Figure 1.2-14: Hart-Smith's shear stress distribution.</td><td>39</td></tr><tr><td>Figure 1.2-15: Relation between the overlap length and the adhesive shear stress distributions for Hart-Smith model, for a) short overlap and b) long overlap.</td><td>39</td></tr><tr><td>Figure 1.2-16: Scheme presenting the relationship between the ultimate capacity and the bonded length.</td><td>40</td></tr><tr><td>Figure 1.2-17: Bonded joint possible singularities.</td><td>41</td></tr><tr><td>Figure 1.2-18: Schemes of the M4 approaches.</td><td>43</td></tr><tr><td>Figure 1.3-1: The three modes of loading of a crack.</td><td>45</td></tr><tr><td>Figure 1.3-2: Energy released with the crack propagation.</td><td>46</td></tr><tr><td>Figure 1.3-3: a) Cracked plate at initial state, b) Cracked plate after fixed displacement Δ or fixed loading P.</td><td>47</td></tr><tr><td>Figure 1.3-4: J-integral contour path Γ around a crack tip in two dimensions.</td><td>48</td></tr><tr><td>Figure 1.3-5: Typical analytical stress state and stress criteria curve.</td><td>50</td></tr><tr><td>Figure 1.3-6: Energy release rate and energy criteria along the bonded length.</td><td>51</td></tr><tr><td>Figure 1.3-7: Two-steps VCCT overview.</td><td>52</td></tr><tr><td>Figure 1.3-8: Coupled stress-energy criterion at critical load level (the extent of the damaged zone is equal to the minimum propagation of the crack).</td><td>52</td></tr><tr><td>Figure 1.4-1: Representation of the fracture process zone and a bilinear cohesive zone law for mode I.</td><td>53</td></tr><tr><td>Figure 1.4-2: Four-node linear cohesive elements separation in FEM.</td><td>54</td></tr><tr><td>Figure 1.4-3: Models: a) with thick cohesive zone model representing the adhesive layer between the adherents, b) with zero thickness cohesive elements with adhesive volume elements.</td><td>55</td></tr><tr><td>Figure 1.4-4: Dugdale cohesive law.</td><td>56</td></tr></table>
<table>
  <tr>
    <td>Figure 1.4-5: Needleman cohesive law</td>
    <td>56</td>
  </tr>
  <tr>
    <td>Figure 1.4-6: Trapezoidal cohesive law</td>
    <td>57</td>
  </tr>
  <tr>
    <td>Figure 1.4-7: Bilinear cohesive law</td>
    <td>57</td>
  </tr>
  <tr>
    <td>Figure 1.4-8: Peterson cohesive law</td>
    <td>57</td>
  </tr>
  <tr>
    <td>Figure 1.4-9: Mixed mode cohesive law surface (Power law criterion, [83])</td>
    <td>59</td>
  </tr>
  <tr>
    <td>Figure 1.4-10: DCB test ([87])</td>
    <td>60</td>
  </tr>
  <tr>
    <td>Figure 1.4-11: Exemple of DCB P(δ) curve</td>
    <td>60</td>
  </tr>
  <tr>
    <td>Figure 1.4-12: Typical DCB C(a3) curve</td>
    <td>60</td>
  </tr>
  <tr>
    <td>Figure 1.4-13: G(a) curve</td>
    <td>62</td>
  </tr>
  <tr>
    <td>Figure 1.4-14: ENF sample</td>
    <td>62</td>
  </tr>
  <tr>
    <td>Figure 1.4-15: Exemple of ENF P(δ) curve</td>
    <td>63</td>
  </tr>
  <tr>
    <td>Figure 1.4-16: MMB test bed and sample</td>
    <td>63</td>
  </tr>
  <tr>
    <td>Figure 1.4-17: MMB Test configuration</td>
    <td>64</td>
  </tr>
  <tr>
    <td>Figure 1.4-18: MMB exemple P(δ) curve</td>
    <td>65</td>
  </tr>
  <tr>
    <td>Figure 1.4-19: Blister test sample</td>
    <td>65</td>
  </tr>
  <tr>
    <td>Figure 1.4-20: 4ENF sample</td>
    <td>66</td>
  </tr>
  <tr>
    <td>Figure 1.4-21: ELS sample</td>
    <td>66</td>
  </tr>
  <tr>
    <td>Figure 1.4-22: Single lap test</td>
    <td>66</td>
  </tr>
  <tr>
    <td>Figure 1.4-23: Spelt test</td>
    <td>67</td>
  </tr>
  <tr>
    <td>Figure 1.4-24: DCB-UBM test frame</td>
    <td>67</td>
  </tr>
  <tr>
    <td>Figure 1.4-25: Arcan test</td>
    <td>68</td>
  </tr>
  <tr>
    <td>Figure 2.2-1: Scheme representing strain distributions for two characteristic cross-sections into the ENF sample during loading</td>
    <td>73</td>
  </tr>
  <tr>
    <td>Figure 2.2-2: Strain into the DCB sample during loading</td>
    <td>74</td>
  </tr>
  <tr>
    <td>The MMB test can be analyzed as the sum of an ENF and a DCB test. Upstream from the crack front (on the right in Figure 2.2-3), the MMB sample corresponds to a beam of thickness 2hs + ha while downstream from the crack front (on the left in Figure 2.2-4), the sample is equivalent to the superposition of two beams of thickness hs</td>
    <td>74</td>
  </tr>
  <tr>
    <td>Figure 2.2-5: Theoretical outer strain profiles on the lower and upper surface of an MMB sample</td>
    <td>75</td>
  </tr>
  <tr>
    <td>Figure 2.3-1: Photo of the DCB sample and DCB test bed</td>
    <td>77</td>
  </tr>
  <tr>
    <td>Figure 2.3-2: Photo of the ENF sample and ENF test bed</td>
    <td>78</td>
  </tr>
  <tr>
    <td>Figure 2.3-3: Photo of the MMB sample and MMB test bed</td>
    <td>78</td>
  </tr>
  <tr>
    <td>Figure 2.3-4: Tested positions for the optical fiber on the sample (flank, transverse, embedded and on the top)</td>
    <td>79</td>
  </tr>
  <tr>
    <td>Figure 2.3-5: a) Example of optical fiber application on a specimen: option 3, b) DCB specimen with surface bonded optical fiber with Araldite-2014-1</td>
    <td>79</td>
  </tr>
  <tr>
    <td>Figure 2.3-6: Scheme of the two studied optical fiber technologies and connection to the samples</td>
    <td>80</td>
  </tr>
  <tr>
    <td>Figure 2.3-7: Local strain measurement during an ENF test with polyimide and acrylate optical fiber coating</td>
    <td>80</td>
  </tr>
  <tr>
    <td>Figure 2.4-1: DCB resin A load/time curve</td>
    <td>81</td>
  </tr>
  <tr>
    <td>Figure 2.4-2: DCB exterior optical fiber continuous strain measurements along lap length for different times</td>
    <td>82</td>
  </tr>
  <tr>
    <td>Figure 2.4-3: Crack propagation curve for the DCB test - data processing via local mean approximation</td>
    <td>82</td>
  </tr>
  <tr>
    <td>Figure 2.4-4: Visual measurement of the crack length of the DCB test, a) in the middle and b) at the border</td>
    <td>83</td>
  </tr>
  <tr>
    <td>Figure 2.5-1: ENF resin A load / time curve, a) test 1 and b) test 2</td>
    <td>84</td>
  </tr>
</table>
<table>
  <tr>
    <td>Figure 2.5-2: Exterior optical fiber distributed strain measurements between times 154.76 s and 203.0 s.</td>
    <td>85</td>
  </tr>
  <tr>
    <td>Figure 2.5-3: Shift of the first peak of the strain profiles measured during the ENF test.</td>
    <td>85</td>
  </tr>
  <tr>
    <td>Figure 2.5-4: Crack propagation curve for the ENF test – raw data processing via local mean approximation.</td>
    <td>86</td>
  </tr>
  <tr>
    <td>Figure 2.5-5: Visual measurement of the crack length at the end of the ENF test.</td>
    <td>86</td>
  </tr>
  <tr>
    <td>Figure 2.5-6: Embedded distributed optical fiber strain measurements before crack propagation.</td>
    <td>87</td>
  </tr>
  <tr>
    <td>Figure 2.5-7: Embedded distributed optical fiber strain measurements after crack propagation.</td>
    <td>88</td>
  </tr>
  <tr>
    <td>Figure 2.5-8: Finite element mesh and strain measurement area.</td>
    <td>88</td>
  </tr>
  <tr>
    <td>Figure 2.5-9: Comparison of strain profiles along lap length: elastic FEM analysis vs experimental measurements at 39 s, a) exterior measurement, b) embedded measurement.</td>
    <td>89</td>
  </tr>
  <tr>
    <td>Figure 2.5-10: Comparison of strain profiles along lap length: elastic FEM analysis vs experimental measurements at t<sub>B</sub>= 58 s, a) exterior measurement, b) embedded measurement.</td>
    <td>90</td>
  </tr>
  <tr>
    <td>Figure 2.6-1: MMB test load/displacement curve.</td>
    <td>90</td>
  </tr>
  <tr>
    <td>Figure 2.6-2: Optical fiber continuous strain measurements between time 100s and 174s.</td>
    <td>91</td>
  </tr>
  <tr>
    <td>Figure 2.6-3: Crack propagation curve for the MMB test.</td>
    <td>91</td>
  </tr>
  <tr>
    <td>Figure 2.6-4: a) Visual measurements of the crack length at the end of the MMB test, a) at mid width, b) on the border.</td>
    <td>92</td>
  </tr>
  <tr>
    <td>Figure 3.1-1: a) Primer application with the lint free cloth, b) Primer applied in thin layer.</td>
    <td>96</td>
  </tr>
  <tr>
    <td>Figure 3.1-2: ENF Samples bonding.</td>
    <td>96</td>
  </tr>
  <tr>
    <td>Figure 3.1-3: Resin flow during the bonding process.</td>
    <td>97</td>
  </tr>
  <tr>
    <td>Figure 3.1-4: Example of speckle and initial crack front.</td>
    <td>97</td>
  </tr>
  <tr>
    <td>Figure 3.2-1: DCB resin B fracture surface.</td>
    <td>98</td>
  </tr>
  <tr>
    <td>Figure 3.2-2: Resin B fracture surface for different modes.</td>
    <td>98</td>
  </tr>
  <tr>
    <td>Figure 3.2-3: DCB tests load/displacement curves.</td>
    <td>99</td>
  </tr>
  <tr>
    <td>Figure 3.2-4: a) II-I test load/time curve, and b) crack propagation curve.</td>
    <td>100</td>
  </tr>
  <tr>
    <td>Figure 3.2-5: a) Linear approximation of the C<sup>1/3</sup>=f(a) curve for the Δ computation, b) Linear approximation of Ln(C)=f(Ln(a)) for n computation.</td>
    <td>100</td>
  </tr>
  <tr>
    <td>Figure 3.2-6: Critical toughness in mode I at initiation and stabilization for the II-1 test.</td>
    <td>101</td>
  </tr>
  <tr>
    <td>Figure 3.2-7: Critical toughness computation results in mode I.</td>
    <td>102</td>
  </tr>
  <tr>
    <td>Figure 3.2-8: ENF tests load/displacement curves with primer.</td>
    <td>103</td>
  </tr>
  <tr>
    <td>Figure 3.2-9: ENF tests load/displacement curves without primer.</td>
    <td>103</td>
  </tr>
  <tr>
    <td>Figure 3.2-10: Linear approximation of the C=f(a<sup>3</sup>) curve with primer.</td>
    <td>104</td>
  </tr>
  <tr>
    <td>Figure 3.2-11: Linear approximation of the C=f(a<sup>3</sup>) curve without primer.</td>
    <td>104</td>
  </tr>
  <tr>
    <td>Figure 3.2-12: ASTM Critical toughness computation.</td>
    <td>105</td>
  </tr>
  <tr>
    <td>Figure 3.2-13: MMB load/displacement curves.</td>
    <td>105</td>
  </tr>
  <tr>
    <td>Figure 3.2-14: Critical toughness average value in function of the mode II ratio for the resin B.</td>
    <td>106</td>
  </tr>
  <tr>
    <td>Figure 3.3-1: Adhesive joint and J-integral integration path.</td>
    <td>107</td>
  </tr>
  <tr>
    <td>Figure 3.3-2: ENF J-integral integration path A+B+C+D.</td>
    <td>107</td>
  </tr>
  <tr>
    <td>Figure 3.3-3: ENF local sliding at the crack front.</td>
    <td>109</td>
  </tr>
  <tr>
    <td>Figure 3.3-4: DCB J-integral integration path A+B+C.</td>
    <td>109</td>
  </tr>
  <tr>
    <td>Figure 3.3-5: Cohesive zone ahead of the crack tip.</td>
    <td>110</td>
  </tr>
  <tr>
    <td>Figure 3.3-6: DIC measurement point on test samples.</td>
    <td>111</td>
  </tr>
  <tr>
    <td>Figure 3.3-7: ENF test DIC speckle showing difficulties to have a precise initial crack front position based on the marking.</td>
    <td>112</td>
  </tr>
</table>
<table>
  <tr>
    <td>Figure 3.3-8: DCB test DIC X displacement map.</td>
    <td>112</td>
  </tr>
  <tr>
    <td>Figure 3.3-9: DCB test DIC X displacement profiles through the sample thickness.</td>
    <td>112</td>
  </tr>
  <tr>
    <td>Figure 3.3-10: Example of ENF test DIC X displacement map and local profiles.</td>
    <td>113</td>
  </tr>
  <tr>
    <td>Figure 3.3-11: ENF test DIC X displacement profiles through sample thickness at the crack front position.</td>
    <td>113</td>
  </tr>
  <tr>
    <td>Figure 3.3-12: ENF SV-1 test, J-integral computation depending of the error of position of the X displacement profile.</td>
    <td>114</td>
  </tr>
  <tr>
    <td>Figure 3.3-13: Impact of the position of the profile on the Jc computation for ENF tests.</td>
    <td>114</td>
  </tr>
  <tr>
    <td>Figure 3.3-14: Impact of the position of the profile on the Jc computation for DCB tests.</td>
    <td>115</td>
  </tr>
  <tr>
    <td>Figure 3.3-15: J-integral computation curves for the DCB tests.</td>
    <td>115</td>
  </tr>
  <tr>
    <td>Figure 3.3-16: Cohesive laws in mode I based on the J-integral curves derivation.</td>
    <td>116</td>
  </tr>
  <tr>
    <td>Figure 3.3-17: J-integral computation curves for the ENF tests without primer.</td>
    <td>116</td>
  </tr>
  <tr>
    <td>Figure 3.3-18: J-integral computation curves for the ENF tests with primer.</td>
    <td>117</td>
  </tr>
  <tr>
    <td>Figure 3.3-19: Cohesive law in mode II based on the J-integral curves derivation.</td>
    <td>118</td>
  </tr>
  <tr>
    <td>Figure 4.1-1: Graphic showing all the parameters defining the bilinear cohesive law in mode I.</td>
    <td>123</td>
  </tr>
  <tr>
    <td>Figure 4.1-2: Cohesive element implementation on the bonded area, contact definition of the initially not bonded area.</td>
    <td>124</td>
  </tr>
  <tr>
    <td>Figure 4.1-3: DCB model representing the adhesive layer with cohesive elements only.</td>
    <td>124</td>
  </tr>
  <tr>
    <td>Figure 4.1-4: ENF model representing the adhesive layer with cohesive elements only.</td>
    <td>124</td>
  </tr>
  <tr>
    <td>Figure 4.1-5: MMB model representing the adhesive layer with cohesive elements only.</td>
    <td>125</td>
  </tr>
  <tr>
    <td>Figure 4.1-6: Direct method flowchart.</td>
    <td>126</td>
  </tr>
  <tr>
    <td>Figure 4.2-1: Indirect method flowchart.</td>
    <td>128</td>
  </tr>
  <tr>
    <td>Figure 4.2-2: Mesh of an ENF for an element size of 1 mm.</td>
    <td>129</td>
  </tr>
  <tr>
    <td>Figure 4.2-3: Load/displacement curves for different mesh sizes for the ENF model with an initial crack length of 74 mm.</td>
    <td>130</td>
  </tr>
  <tr>
    <td>Figure 4.2-4: Comparison between experimental and numerical load/displacement curves for different cohesive stiffness for ENF models.</td>
    <td>130</td>
  </tr>
  <tr>
    <td>Figure 4.2-5: Comparison between experimental (a₀ = 74 mm) and numerical load/displacement curves for different critical shear stress for ENF models.</td>
    <td>131</td>
  </tr>
  <tr>
    <td>Figure 4.2-6: Cohesive parameters in mode II obtained for indirect method for resin B steel-to-steel bonding.</td>
    <td>132</td>
  </tr>
  <tr>
    <td>Figure 4.2-7: Comparison of experimental and numerical load/displacement curves obtained for ENF tests and models, with initial crack lengths of 59 mm and 90 mm.</td>
    <td>132</td>
  </tr>
  <tr>
    <td>Figure 4.2-8: 3D Model of the ENF (the contact zones are highlighted).</td>
    <td>133</td>
  </tr>
  <tr>
    <td>Figure 4.2-9: Load/displacement curves for the 2D and 3D model of the ENF test.</td>
    <td>134</td>
  </tr>
  <tr>
    <td>Figure 4.2-10: Plot of the cohesive element damage during the ENF test showing a planar crack propagation.</td>
    <td>134</td>
  </tr>
  <tr>
    <td>Figure 4.2-11: Crack front observations on an ENF sample at the end of the test.</td>
    <td>134</td>
  </tr>
  <tr>
    <td>Figure 4.2-12: Load/displacement curves for different mesh sizes for the DCB model.</td>
    <td>135</td>
  </tr>
  <tr>
    <td>Figure 4.2-13: Numerical load/displacement curves for different cohesive stiffness for DCB models.</td>
    <td>136</td>
  </tr>
  <tr>
    <td>Figure 4.2-14: Comparison of load/displacement curves for different critical peel stresses for DCB models.</td>
    <td>137</td>
  </tr>
  <tr>
    <td>Figure 4.2-15: Comparison of experimental and numerical load/displacement curves of DCB resin B.</td>
    <td>138</td>
  </tr>
  <tr>
    <td>Figure 4.2-16: Resulting indirect cohesive law in mode I for steel-to-steel bonded assembly with resin B.</td>
    <td>138</td>
  </tr>
  <tr>
    <td>Figure 4.2-17: 3D DCB model.</td>
    <td>139</td>
  </tr>
  <tr>
    <td>Figure 4.2-18: Comparison of load/displacement curves for DCB model in 2D and 3D with a 2mm mesh size.</td>
    <td>139</td>
  </tr>
</table>
<table>
  <tbody>
    <tr>
      <td>Figure 4.2-19: Fracture surface along the width for the 3D DCB model.</td>
      <td>140</td>
    </tr>
    <tr>
      <td>Figure 4.2-20: Indirect cohesive laws obtained in mode I and mode II.</td>
      <td>140</td>
    </tr>
    <tr>
      <td>Figure 4.2-21: Comparison of load/displacement curves for MMB experimental test and model with a 1mm mesh size, a mixed mode ratio of 0.7 and different values of the coupling parameter α.</td>
      <td>141</td>
    </tr>
    <tr>
      <td>Figure 4.2-22: Comparison of load/displacement curves for MMB experimental test and model with a 1mm mesh size, a mixed mode ratio of 0.5 and different values of the coupling parameter α.</td>
      <td>141</td>
    </tr>
    <tr>
      <td>Figure 4.3-1: Experimental and proposed numerical cohesive laws in pure Mode II for ENF test.</td>
      <td>143</td>
    </tr>
    <tr>
      <td>Figure 4.3-2: Experimental and proposed numerical cohesive laws in pure Mode I for DCB test.</td>
      <td>143</td>
    </tr>
    <tr>
      <td>Figure 4.3-3: Numerical and experimental ENF load/displacement curves for different initial crack lengths.</td>
      <td>144</td>
    </tr>
    <tr>
      <td>Figure 4.3-4: Numerical direct method and experimental DCB load/displacement curves.</td>
      <td>144</td>
    </tr>
    <tr>
      <td>Figure 4.3-5: Comparison of numerical and experimental load/displacement curves for MMB test with 50% of mode II, for different values of the coupling parameter, using direct cohesive law.</td>
      <td>145</td>
    </tr>
    <tr>
      <td>Figure 4.3-6: Comparison of numerical and experimental load/displacement curves for MMB test with 70% of mode II, for different values of the coupling parameter values, using direct cohesive law.</td>
      <td>146</td>
    </tr>
    <tr>
      <td>Figure 4.4-1: Final DCB cohesive laws.</td>
      <td>147</td>
    </tr>
    <tr>
      <td>Figure 4.4-2: Final ENF cohesive laws.</td>
      <td>148</td>
    </tr>
    <tr>
      <td>Figure 5.1-1: Full-size test geometries for tension and bending/tension loadings.</td>
      <td>149</td>
    </tr>
    <tr>
      <td>Figure 5.1-2: Full size tension and bending samples instrumentation.</td>
      <td>150</td>
    </tr>
    <tr>
      <td>Figure 5.1-3: a) Initial tension test without instrumentation, b) Tension sample lower end with optical fiber and gauge instrumentation.</td>
      <td>151</td>
    </tr>
    <tr>
      <td>Figure 5.1-4: Bending test with instrumentation.</td>
      <td>152</td>
    </tr>
    <tr>
      <td>Figure 5.1-5: Adhesive failure and crack front shape of the samples.</td>
      <td>152</td>
    </tr>
    <tr>
      <td>Figure 5.1-6: Tension test load/displacement curves.</td>
      <td>153</td>
    </tr>
    <tr>
      <td>Figure 5.1-7: Gauges 3 and 4 strain measurement for the three samples in tension.</td>
      <td>154</td>
    </tr>
    <tr>
      <td>Figure 5.1-8: Load/displacement curves of the bending tests.</td>
      <td>155</td>
    </tr>
    <tr>
      <td>Figure 5.1-9: Gauges 3 and 4 strain measurements during the bending tests.</td>
      <td>155</td>
    </tr>
    <tr>
      <td>Figure 5.2-1: Stress- energy coupled criteria.</td>
      <td>156</td>
    </tr>
    <tr>
      <td>Figure 5.2-2: Finite element model boundary conditions for tension test.</td>
      <td>157</td>
    </tr>
    <tr>
      <td>Figure 5.2-3: Finite element model boundary conditions for bending test.</td>
      <td>157</td>
    </tr>
    <tr>
      <td>Figure 5.2-4: Stress analysis meshing.</td>
      <td>157</td>
    </tr>
    <tr>
      <td>Figure 5.2-5: Mesh sensitivity analysis for the tension test: shear stress.</td>
      <td>158</td>
    </tr>
    <tr>
      <td>Figure 5.2-6: Mesh sensitivity analysis for the tension test: peel stress.</td>
      <td>158</td>
    </tr>
    <tr>
      <td>Figure 5.2-7: Mesh sensitivity analysis for the tension test: principal stress.</td>
      <td>159</td>
    </tr>
    <tr>
      <td>Figure 5.2-8: Mesh sensitivity analysis for the bending test: shear stress.</td>
      <td>159</td>
    </tr>
    <tr>
      <td>Figure 5.2-9: Tension and bending model shear stress.</td>
      <td>160</td>
    </tr>
    <tr>
      <td>Figure 5.2-10: Illustration of the energy criteria with a critical interface ERR Gc and a characteristic length ac.</td>
      <td>160</td>
    </tr>
    <tr>
      <td>Figure 5.2-11: Example of the definition of the critical ERR with the B-K criterion for a resin C.</td>
      <td>161</td>
    </tr>
    <tr>
      <td>Figure 5.2-12: Resin B fracture test results and B-K criterion fitting.</td>
      <td>162</td>
    </tr>
    <tr>
      <td>Figure 5.2-13: VCCT energy release rate for the tension model.</td>
      <td>162</td>
    </tr>
    <tr>
      <td>Figure 5.2-14: VCCT energy release rate for the bending model.</td>
      <td>162</td>
    </tr>
    <tr>
      <td>Figure 5.2-15: VCCT G<sub>II</sub>/G<sub>tot</sub> mode ratio for the tension and bending model.</td>
      <td>163</td>
    </tr>
  </tbody>
</table>
<table>
  <tbody>
    <tr>
      <td>Figure 5.2-16: Tension test coupled stress-energy criterion computation for: a) experimental failure load, b) 1.15 * experimental failure load, c) 1.22 * experimental failure load and d) 1.4 * experimental failure load.</td>
      <td>164</td>
    </tr>
    <tr>
      <td>Figure 5.2-17: Bending test coupled stress-energy criterion computation for: a) experimental failure load and b) 1.13 * experimental failure load.</td>
      <td>164</td>
    </tr>
    <tr>
      <td>Figure 5.3-1: Cohesive model meshing.</td>
      <td>165</td>
    </tr>
    <tr>
      <td>Figure 5.3-2: a) Bending test load/displacement curves, b) Tension test load/displacement curves.</td>
      <td>166</td>
    </tr>
    <tr>
      <td>Figure 5.3-3: Experimental and numerical gauge 1 measurements for the tension tests.</td>
      <td>167</td>
    </tr>
    <tr>
      <td>Figure 5.3-4: Zoom on the non-linear part of the curves for gauge 1 measurements for the tension tests.</td>
      <td>168</td>
    </tr>
    <tr>
      <td>Figure 5.3-5: Experimental and numerical gauge 5 data for the tension test.</td>
      <td>168</td>
    </tr>
    <tr>
      <td>Figure 5.3-6: Experimental and numerical gauge 3 measurements for the tension test.</td>
      <td>169</td>
    </tr>
    <tr>
      <td>Figure 5.3-7: Experimental optical fiber strain measurements for tension test n°3 and numerical strain profile results.</td>
      <td>170</td>
    </tr>
    <tr>
      <td>Figure 5.3-8: Experimental and numerical gauge 1 measurements for the bending tests.</td>
      <td>171</td>
    </tr>
    <tr>
      <td>Figure 5.3-9: Experimental and numerical gauge 5 measurements for the bending tests.</td>
      <td>172</td>
    </tr>
    <tr>
      <td>Figure 5.3-10: Experimental and numerical gauge 3 measurements for the bending tests.</td>
      <td>173</td>
    </tr>
    <tr>
      <td>Figure 5.3-11: Experimental optical fiber strain measurements for the bending test n°2 and numerical strain profile results.</td>
      <td>174</td>
    </tr>
    <tr>
      <td>Figure 5.3-12: Experimental crack front shape and FE method crack front shape prediction.</td>
      <td>175</td>
    </tr>
    <tr>
      <td>Figure 5.4-1: Normalized failure load predictions of direct, indirect and coupled stress-energy criteria methods compared to experimental results.</td>
      <td>176</td>
    </tr>
  </tbody>
</table>
Table of tables

<table><tr><td>Table 2.1-1: Comparison (advantage and limitations) of several techniques for crack front measurement.</td><td>72</td></tr><tr><td>Table 2.3-1: Resin A mechanical properties in tension.</td><td>76</td></tr><tr><td>Table 2.7-1: Crack length from the experimental investigations using exterior optical fiber and visual inspection.</td><td>92</td></tr><tr><td>Table 3.1-1: Resin B mechanical properties in tension.</td><td>95</td></tr><tr><td>Table 3.1-2: Fracture tests nomenclature.</td><td>96</td></tr><tr><td>Table 3.2-1: DCB test nomenclature and crack propagation information.</td><td>99</td></tr><tr><td>Table 3.2-2: Standard critical toughness characterization with optical fiber crack length measurement.</td><td>101</td></tr><tr><td>Table 3.2-3: ENF test nomenclature.</td><td>102</td></tr><tr><td>Table 3.2-4: Critical toughness computation following the ASTM D7905 standard.</td><td>104</td></tr><tr><td>Table 3.2-5: Critical toughness computation of the mixed mode test.</td><td>106</td></tr><tr><td>Table 3.3-1: DCB test J-integral polynomial approximation.</td><td>115</td></tr><tr><td>Table 3.3-2: Critical toughness computation in mode I based on the J-integral curves.</td><td>116</td></tr><tr><td>Table 3.3-3: ENF test J-integral polynomial approximation.</td><td>117</td></tr><tr><td>Table 3.3-4: Critical toughness computation in mode II based on the J-integral curves.</td><td>117</td></tr><tr><td>Table 3.4-1: Critical toughness computation in mode I.</td><td>118</td></tr><tr><td>Table 3.4-2: Critical toughness computation in mode II.</td><td>119</td></tr><tr><td>Table 4.2-1: Comparison of experimental and numerical failure forces for different critical shear stress for ENF model with an initial crack length of 74mm.</td><td>131</td></tr><tr><td>Table 4.2-2: Cohesive parameters in mode II obtained for indirect method for Resin B steel-steel for indirect method.</td><td>131</td></tr><tr><td>Table 4.2-3: Comparison of failure forces obtained numerically and experimentally for ENF tests and models with different initial crack lengths.</td><td>132</td></tr><tr><td>Table 4.2-4: Failure forces for different mesh sizes for DCB model.</td><td>135</td></tr><tr><td>Table 4.2-5: Comparison of experimental and numerical failure forces for different critical peel stresses for DCB models.</td><td>137</td></tr><tr><td>Table 4.2-6: Cohesive parameters in mode I for indirect method.</td><td>138</td></tr><tr><td>Table 4.2-7: Comparison of experimental and numerical failure loads for MMB test for a mode mixity of 50% and 70% and for different coupling parameter values.</td><td>142</td></tr><tr><td>Table 4.3-1: Cohesive parameters using direct method for mode I and II.</td><td>142</td></tr><tr><td>Table 4.3-2: Comparison of failure loads in mode II obtained numerically by the direct method and experimentally by ENF tests.</td><td>144</td></tr><tr><td>Table 4.3-3: Comparison of numerical and experimental failure loads for the DCB direct method.</td><td>145</td></tr><tr><td>Table 4.3-4: Comparison of experimental and numerical failure loads for MMB test with 50% of mode II, for different values of the coupling parameter, using direct cohesive law.</td><td>145</td></tr><tr><td>Table 4.3-5: Comparison of experimental and numerical failure loads for MMB test 70% of mode II, for different values of the coupling parameter, using direct cohesive law.</td><td>146</td></tr><tr><td>Table 4.3-6: Resulting parameters of cohesive laws using direct method.</td><td>146</td></tr><tr><td>Table 5.1-1: Strain and load at failure for the tension test.</td><td>154</td></tr></table>
<table><tr><td>Table 5.1-2: Strain and load at failure of the bending tests.</td><td>155</td></tr><tr><td>Table 5.3-1: Resulting parameters of cohesive law for resin B steel-to-steel assembly characterization using direct method.</td><td>166</td></tr><tr><td>Table 5.3-2: Resulting parameters of cohesive law for resin B steel-to-steel assembly characterization using indirect method.</td><td>166</td></tr><tr><td>Table 5.3-3: Several characteristic values determined through experimental and numerical analyses using the developed cohesive zone approaches (direct and indirect methods) for gauges 1 and 5, for the tension tests.</td><td>169</td></tr><tr><td>Table 5.3-4: Several characteristic values determined through experimental and numerical analyses using the developed cohesive zone approaches (direct and indirect methods) for gauge 3, for the tension tests.</td><td>170</td></tr><tr><td>Table 5.3-5: Several characteristic values determined through experimental and numerical analyses using the developed cohesive zone approaches (direct and indirect methods) for gauge 1 and 5, for the bending tests.</td><td>172</td></tr><tr><td>Table 5.3-6: Several characteristic values determined through experimental and numerical analyses using the developed cohesive zone approaches (direct and indirect methods) for gauge 3, for the bending tests.</td><td>173</td></tr><tr><td>Table 5.4-1: Failure load prediction of direct, indirect methodologies and coupled stress-energy criteria compared to experimental results for real-scale investigations.</td><td>176</td></tr></table>
# General Introduction

The maintenance of marine steel structures installed in harsh offshore environment (like tropical areas) is a great challenge. Vessels and mobile offshore units can be maintained and repaired onshore in shipyards, units such as Floating Production Storage and Offloading platforms (FPSO) are permanently moored at sea and shall be maintained on site. Similarly, naval ships may also be faced with the need to perform repairs in short period of time at sea to maintain operability. On FPSO units, the corrosion is a permanent threat due to high temperature and high humidity conditions. Bonded repair solutions which present several advantages (short down-time and non-intrusively process) are actually in development. However, repairing corroded areas on large marine structures by bonding patch imposes important constraints on the design. The patch lies in fully stressed area (for instance area subject to hull girder flexibility), causing high stresses in the bond line edges. Thus, a fine apprehension of adhesion mechanics and strength is critical for designing highly reliable patch repairs.

This study is part of a project called Strength Bond aiming at developing a bonded composite repair for corroded steel plate for offshore applications. The development of a composite repair in such conditions seems particularly appropriate for various reasons. Composite materials show high performance, can be easily adapted to various 3-dimensional shape, and are easy to transport and install. In addition, the use of adhesive bonding is particularly adapted as it limits local stress concentrations, and may avoid operations at high temperatures. Yet, the lack of commonly adopted design approach for adhesively bonded joints still limits the use of such techniques.

One of the main issues encountered in bonded assemblies is the risk of brittle failure. Consequently, due to safety considerations, it is often required that adhesively bonded structures (particularly those employed in primary load/bearing applications) include mechanical fasteners to limit this type of failure (welding, bolts, ...). These practices generally result in heavier, more expensive and less efficient assembly. Moreover, this is not relevant in our application case, as it would require « hot works » that may damage the existing structure, raise safety issues or limit the operation during the repair/reinforcement application.

Better understanding of the failure of bonded joint and the development of reliable design methodologies are thus necessary to increase confidence and to lead to more efficient application of composite bonded repairs in offshore applications. It is also a necessary step for further damage tolerance assessment and ageing assessment. It was therefore decided to focus in this work on adhesively bonded joints design.

When designing adhesively bonded joints, several steps are generally required. The first step aims at assessing the adhesive or the joints capacities through standardized experimental investigations. It allows obtaining failure criteria data and sometimes mechanical behavior information. It may also give information related to surface preparation and failure modes. The second step requires having access to mechanical analysis tools that may be analytical or numerical. Their use provides information related to design predictions. Comparison of both results (failure criteria and design predictions) is then carried out, and, if needed, design is modified.

Several design approaches have been developed for adhesively bonded joints (relying on continuous mechanics, fracture mechanics or both. This work intends to develop a methodology
assessing at each step (characterization and design assessment) the precision of the obtained
results and the sensitivity of the approach.

First chapter will be dedicated to a literature review on adhesively bonded joints and their
design. After general information related to the specific case of bonded joints, and the issue of
adhesion, main existing design strategies (stresses and energy approaches) are developed giving
for each the advantages and the drawbacks. A focus is then done on the use of the cohesive
zone approach that was chosen for the developed methodology. The associated standard
fracture mechanics experimental investigations and critical toughness computation methods are
also described.

The second chapter will intend at improving the experimental characterization through fracture mechanics investigations. In order to improve the robustness and precision of the measurement, it was decided to study if the use of continuous optical fiber measurement for crack monitoring could be suitable. Several experimental investigations are presented in mode I, II and mixed mode. Obtained optical fiber crack length measurement are compared to visual and numerical measurement and a precise methodology for the obtention of the critical toughnesses is proposed and assessed.

The third chapter presents the application of this methodology to a wider experimental campaign. Both standards and J-integral approaches are carried out in order to determine the critical toughnesses and, for the J-integral approach, the cohesive zone model. The obtained results are commented and compared.

Chapter fourth is dedicated to the presentation of the modeling strategy relying on the use of
finite element models. The cohesive elements are presented and different strategies aiming at
obtaining the cohesive zone models are also presented. Two main methods are being considered
and compared for mode I and II: the direct and the indirect methods. The mixed mode case is
also deeply studied.

The last chapter intends at applying the developed methodology on real scale adhesively bonded
joints, using the improved mechanical investigations, the obtained critical toughnesses and the
cohesive zone models using the different strategies. The experimental results, load at failure,
local and general samples behavior are compared to design predictions. In order to assess the
precision of the methodology, an alternative design methodology is carried out (the coupled
stress energy approach). Conclusion are express on the efficiency and robustness of each
methodology to predict behavior of bonded patch.
# Chapter 1: State of the art regarding the design of adhesively bonded joints

In the first part, some generalities regarding the use of structural adhesive bonding will be described. This will allow to address main issues related to this assembly technique and the corresponding glossary. In the second part, the different strategies available to determine the adhesively bonded joints mechanical state will be presented. At first, main analytical developments will be detailed. Then, different numerical strategies will be described. In the third part, the different failure criteria that were developed and used for the considered application will be listed. A classification is proposed and allows distinguishing the classical criteria used in continuum mechanics, the fracture mechanics criteria and the mixed criteria. Finally, the cohesive zone model approach, comprising a modeling strategy and failure criteria, will be described. The main methodologies used to determine the parameters of the cohesive zones will also be highlighted.

## 1.1. Generalities about adhesive bonding

A bonded assembly consists in two adherents (substrates) which are the elements to be assembled, linked by an adhesive which transmits the load from one element to the other.

### 1.1.1. Historical background

The first use of bonded assembly was in 8000 years BC with the use of animal and vegetable adhesives (fish and bones glue) for the assembly of tools and wood structures. Synthetic adhesives were created only around 1850 with the industrialization [1]. During the 20th century, their development is related to the newly born aeronautics industry. The Fokker company started using adhesive bonding in 1915 for wood-to-wood and wood-to-fabric assembly leading the way for other manufacturers [2]. In 1940, adhesive bonds were used on wood spars, on the De Havilland DH 98 Mosquito aircraft. In the 1960s, the aerospace industry also started using adhesive bonding technology, convinced that it was an optimal solution for the manufacture of lightweight and resistant structures. Today, it tends to be more and more extensively used in other areas: sailing, building and civil engineering [3] [4] [5] [6] [7].

### 1.1.2. Adhesion theories

The link between the adherent and the adhesive mainly depends on the adhesion at the interface, which remains a research topic. Adhesion includes all the physico-chemical phenomena that occur when intimate contact is realized between two materials. There is no single model to explain this phenomenon and several theories in different fields have been proposed to explain the adhesion. Main ones are listed here:

- physical adsorption: effect of forces of attraction between atoms and molecules of different nature, acting on the surfaces in contact. The molecules of the adhesive are retained on the surface, thanks to the adsorption forces, which act as soon as the molecules are within $5 \times 10^{-8} \text{cm}$ [8],

- chemical absorption: chemical reaction between the atoms and the molecules of the surfaces in contact [9],
- mechanical anchorage: comes from the creation of geometrical anchorage on rough surfaces. This theory links directly the surface geometry with the resistance of the assembly [10].

Adhesion is therefore closely related to the properties of the surfaces of the two adherents. For this reason, the process of surface preparation prior to adhesive bonding is an important issue. This will determine directly the level of surface wettability and the final level of adhesion forces. Surface energy (or surface tension, or wettability) is a physical quantity used to characterize the state of cohesion of a solid (or liquid). The molecular force of attraction between different materials is related to their surface properties (surface energy and/or surface tension) and conditions their adhesion. For a solid, a high surface energy means a strong molecular attraction, while a low surface energy results in lower forces of attraction. For optimum bonding of two materials, it is necessary to have good wettability (liquid on solid). This results in higher surface energy of the solid (substrate to be bonded) than of the liquid (adhesive) [11].

Either mechanical or chemical surface preparations can be carried out. The most commonly encountered mechanical surface preparation processes are the following:

- sandblasting, that consists in projection of an abrasive powder. The surface may then be covered with a protective film before adhesive bonding (like anti-corrosion film),

- abrasion, that consists in removing the external adherent layer with a specific brush or tool,

- ultrasound treatment, that is an expensive method of cavitating water by overpressure – depression. It cleans the surface without modifying its wettability characteristics.

Regarding the chemical surface preparation, it consists mainly in:

- degreasing with solvents: in that case, derivatives of hydrocarbons or chlorine are generally used. The solvent is chosen according to the contaminating elements,

- using primer agent on the surface prior to bonding to improve chemical compatibility.

### 1.1.3. Failure modes

In the assembly of two materials, the overall resistance depends on the strength of the adherents, the adhesive and the interfaces between adherents and adhesive. Three failure modes of a bonded assembly can be distinguished (Figure 1.1-1):

- cohesive failure, in the adhesive or in the substrates,

- adhesive failure, at the interfaces,

- or a mix of cohesive and adhesive failure.

A cohesive failure is generally preferred to an adhesive failure as it relies on the properties of the materials and not the quantification of the adhesion forces that are strongly related to the surface preparation and the application quality.

<figure><img src="image_9.png" /><figcaption>Figure 1.1-1: Failure modes of an adhesively bonded assembly.</figcaption></figure>
1.1.4. Load and geometries of adhesively bonded joint

Adhesively bonded joints can be submitted to several types of mechanical stresses. Figure 1.1-2 shows the four main types of loads experienced by a bonded assembly: shear, peel, cleavage and traction/compression. An adhesively bonded assembly is generally more efficient in shear and compression [12].

<figure><img src="image_10.png" /><figcaption>Figure 1.1-2: Main types of loads applied on bonded joints.</figcaption></figure>

Depending on the application, various geometries have been proposed for improving the strength (increase shear loading or reduce geometrical singularities) of a bonded assembly. Figure 1.1-3 presents the main encountered geometries of bonded assembly that were studied in the literature. Among those geometries, the butt joint is not well adapted to bonded assembly as it does not induce any shear or compressive stresses. The single lap joint is the most encountered geometry and will therefore be more deeply studied in next part [13].

<figure><img src="image_11.png" /><figcaption>Figure 1.1-3: Most commonly encountered bonded assemblies.</figcaption></figure>
## 1.2. Mechanical state analysis of bonded lap joints

To be able to express and use adapted criteria for the design of the assembly, it is necessary to have access to the internal stress or strain states.

Existing analytical modeling developed for single or double lap joints will be first reviewed. Then an overview of the main numerical approaches available for the designers will be presented.

### 1.2.1. Models based on analytical solutions

It is not possible to exhaustively describe all the proposed analytical models in this part. Only the most recognized and used approaches will be raised with increasing complexity and in order to highlight the main phenomena associated to the design of adhesively bonded joints.

#### 1.2.1.1. Constant shear model

Regarding single lap joints, the simplest analysis that can be applied considers that the shear stress is constant over the entire bonded surface [14]. For that case, only the shear stress is considered and the adherents are considered as rigid (Figure 1.2-1).

<figure><img src="image_12.png" /><figcaption>Figure 1.2-1: Constant shear model.</figcaption></figure>

This rigidity involves the adhesive shear stress $\tau$ to be constant over the overlap length, and it can be expressed by the equation (1).

$$ \tau(x) = \frac{P}{bc} \qquad (1) $$

Where, $P$ is the applied load to the adherent, $b$ is the joint width, and $c$ is the overlap length.

Such hypothesis is only relevant when the overall stiffness of the bonded joint (depending on the thickness, the Young's modulus of the adhesive, width and length of overlap), $k_a$, is low, in comparison with the stiffness of the upper adherent, $k_u$, and lower adherent, $k_l$. These parameters are detailed in equation (2), by unit width.

$$ k_u = \frac{t_u E_u}{c}, k_l = \frac{t_l E_l}{c}, k_a = \frac{G_a c}{t_a} \qquad (2) $$
$E_u$, $E_l$ being the Young modulus of the upper and lower adherent, $G_a$ the shear modulus of the adhesive, and $t_u, t_l, t_a$ the thicknesses of respectively the upper adherent, lower adherent and the adhesive, respectively.

The main advantage of this model is its ease of use. It is used to define the adhesive shear strength in several standards such as TAST (thick adherent shear test), described in ISO 11003-2 [15] and ASTM D 5656 [16] standards (Figure 1.2-2). In these standards, a specific geometry is adopted, in particular thick adherents and short overlap, in order to obtain shear stress profiles as constant as possible. This is however difficult to obtain in the case of structural adhesives (as their stiffness is high) and as there exists actually edge effects.

<figure><img src="image_13.png" /><figcaption>Figure 1.2-2: TAST Test [16].</figcaption></figure>

#### 1.2.1.2. Volkersen analysis or “shear lag” model

The first historical model that considers the flexibility of the adherents was proposed in 1938 by O. Volkersen [17] in 1938, and is called “shear lag” model. Using a 1D model of pure shear, it assumed that the adhesive layer was loaded only in shear and that the adherents followed a purely longitudinal strain. It also assumes that the behavior of the adhesive is linear elastic. The shear stress in the adhesive (Figure 1.2-3) is maximum at $x = 0, x = c$ ($c$ is the overlap length).

<figure><img src="image_14.png" /><figcaption>Figure 1.2-3: Volkersen's model for the single lap joint.</figcaption></figure>

The stress distribution in the adhesive is obtained in that case using the free-body-diagram of an infinitesimal section of the joint (Figure 1.2-4). $\sigma_u$ is the axial stress in the upper adherent, and $\sigma_l$ is the axial stress in the lower adherent.
<figure><img src="image_15.png" /><figcaption>Figure 1.2-4: Infinitesimal free-body-diagram of the Volkersen's model.</figcaption></figure>

Firstly, equilibrium equation for each adherent can be written to obtain a direct relation between the shear stress and tensile stress in the two adherents.

$$ \tau(x) = -t_u \frac{d\sigma_u(x)}{dx} \qquad (3) $$

$$ \tau(x) = t_l \frac{d\sigma_l(x)}{dx} \qquad (4) $$

Following the Hooke equations, the overall shear in the adhesive can also be expressed by equation (5).

$$ \frac{d\tau(x)}{dx} = \frac{G_a}{t_a} \left( -\frac{\sigma_l(x)}{E_l} + \frac{\sigma_u(x)}{E_u} \right) \qquad (5) $$

Using equations (3), (4) and (5), equation (6) can be obtained.

$$ \frac{d^2 \left( -\frac{\sigma_l(x)}{E_l} + \frac{\sigma_u(x)}{E_u} \right)}{dx^2} = \omega^2 \left( -\frac{\sigma_l(x)}{E_l} + \frac{\sigma_u(x)}{E_u} \right) \qquad (6) $$

$$ \text{With} \qquad \omega = \sqrt{\frac{G_a}{t_a} \left( \frac{1}{t_l E_l} + \frac{1}{t_u E_u} \right)} \qquad (7) $$

Based on these equations, an expression of the shear stress distribution along the adhesive layer can be obtained. The unique solution of this equation with the boundary conditions of a single lap joint: $\sigma_l(x = 0) = -\frac{P}{bt_l}$, $\sigma_l(x = c) = 0$, $\sigma_u(x = 0) = 0$, $\sigma_u(x = c) = \frac{P}{bt_u}$ are given in equation (8).

$$ \tau(x) = \frac{G_a}{\omega t_a} \frac{P}{bE_l t_l} \left( \frac{\cosh(\omega x)}{\sinh(\omega c)} \left( \cosh(\omega c) + \frac{k_l}{k_u} \right) - \sinh(\omega x) \right) \qquad (8) $$

This equation (8) can be simplified if the lower and upper adherents have an equivalent rigidity $k_u = k_l$. In that case, the adhesive joint is balanced and equation (9) is obtained.
$$ \tau(x) = \frac{k_a}{\omega k_l c} \frac{P}{bc} \left( \frac{\cosh(\omega x)(1 + \cosh(\omega c))}{\sinh(\omega c)} - \sinh(\omega x) \right) \quad (9) $$

In Figure 1.2-5, the obtained shear stress was plotted for a fixed bondline length of 20 mm and $k_u/k_a$ (representing the stiffness ratio) ranging from 0.005 to 0.5. For the curve corresponding to $k_u/k_a = 0.2$, the part in the middle does not transfer any shear stress. This result illustrates an important phenomenon related to bonding: the existence of a characteristic anchorage length also called effective bond length. The effective bond length corresponds to the length over which the majority of the bond stress is transferred.

<figure><img src="image_16.png" /><figcaption>Figure 1.2-5: Shear stress along the bonded length according to Volkersen's theory for different values of $k_u/k_a$ versus constant shear model.</figcaption></figure>

In Figure 1.2-6, the obtained shear stress according to Volkersen's theory given by equation (8) was plotted for four different bonded lengths: 20 mm, 50 mm, 250 mm and 1 000 mm and for the following geometry and properties: $t_u = t_l = 20$ mm, $t_a = 1$ mm, $G_a = 2$ GPa, $E_u = E_l = 200$ GPa and $F = 1000$ N. We can observe for the curve with $c \ge 250$ mm that any shear stress is transferred in the middle of the bondline. It is the same phenomenon as described in the previous paragraph. In addition, it can be noted that the ultimate shear values are obtained at the edges. For a given applied load level, it can be stated that once the bonded length is greater than the characteristic anchorage length (250mm), those values do not decrease anymore.
<figure><img src="image_17.png" /><figcaption>Figure 1.2-6: Shear stress along the bonded length according to Volkersen's theory for different bonded length versus constant shear model.</figcaption></figure>

The characteristic anchorage length expresses the fact that, in the condition of validity of the model (the adhesive is assumed to have linear elastic behavior), above a long enough length (depending on the material properties), the ultimate strength and the overlap length are independent (Figure 1.2-7). Therefore, the presence of a bond defect outside the area of characteristic anchorage length should not induce a reduction of the ultimate strength of the bonding.

<figure><img src="image_18.png" /><figcaption>Figure 1.2-7: Typical relation between ultimate capacity and overlap length [18].</figcaption></figure>

The model of Volkersen allows an easy understanding of the main phenomenon related to the use of bonded joints. However, it has some problems inducing restrictions in its use. The main issue with the theory of Volkersen is that the eccentricity of the loads is neglected. In that case, bending moments in the adherents is introduced, and thus peel stresses in the adhesive, that may cause premature damage. The Volkersen model cannot predict this phenomenon. Additionally, this model is limited to linear material, whereas a large range of bonded joints uses nonlinear adhesives. Finally, it is interesting to note that the model proposed by Volkersen does not fulfill the generalized beam theory free edge null shear strain fundamental assumption [19].

#### 1.2.1.3. Goland and Reissner approach

To improve the Volkersen model on the stress analysis of bonded joints, Goland & Reissner [20] [21] proposed another approach, based on plates theory, to express the stress state along the bondline for linear elastic material, including the effect of the bending of the adherent resulting in shear and peel stresses along the bonded length (Figure 1.2-8). Their analysis was only developed for the case of balanced joints (similar top and bottom adherends).
<figure><img src="image_19.png" /><figcaption>Figure 1.2-8: Deformed state of a single lap joint in the case of the Goland and Reissner model [22].</figcaption></figure>

The single lap joint problem defined by Goland & Reissner is divided into two different studies:

- the analysis of the bending moment and the rotation of the adherents out of the bonded area, following the plate theory;

- the analysis of the bonded area.

The stress in the assembly of the Goland & Reissner model can be summarized into two free-body-diagram of infinitesimal section of the joint, and adherent (Figure 1.2-9, Figure 1.2-10).

<figure><img src="image_20.png" /><figcaption>Figure 1.2-9: Infinitesimal free-body-diagram of the Goland & Reissner model in the adherent out of the bonded area.</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>Figure 1.2-10: Infinitesimal free-body-diagram of the Goland & Reissner model in the joint.</figcaption></figure>

For the adherent (out of the bonded area) under bending load, two sets of differential equations can be obtained using the plate theory and the Hooke's law stress/strain: equations (10), (11) and (12).

$$ \frac{d^3\tau(x)}{dx^3} = \frac{8G_a t_u}{E t_a} \frac{d\tau(x)}{dx} \qquad (10) $$

$$ \frac{d^2\tau(x)}{dx^2} = \frac{E_a (M_l - M_u)}{t_u R} \qquad (11) $$

$$ \frac{d^4\sigma_y(x)}{dx^4} = -\frac{E_a\sigma_y}{t_u R} \qquad (12) $$
With $M_u$ is the applied bending moment to the upper adherent, $M_l$ is the applied bending moment to the lower adherent, $\sigma_y$ is the normal stress in the adhesive, $t_u$ the thickness of the adherent, $t_a$ the thickness of the adhesive, E the Young modulus of the adherents, $E_a$ the Young modulus of the adhesive, R the flexural rigidity of the adherend and $G_a$ the shear modulus of the adhesive.

The following boundary conditions are considered: equations (13) and (14).

$$ at x = c, M_u = V_u = 0 \text{ and } M_l = M_o, V_l = V_o \qquad (13) $$

$$ at x = 0, M_l = V_l = 0 \text{ and } M_u = -M_o, V_u = V_o \qquad (14) $$

$M_0$ and $V_0$ are respectively the bending moment and the shearing force at the joint edge defined by:

$$ M_0 = k \frac{Ft_u}{2} \qquad (15) $$

$$ \text{and } V_0 = kF \sqrt{\frac{3(1-v^2)F}{t_u E}} \qquad (16) $$

Where $F$ is the applied tensile load per unit width ($F = P/b$) and $k$ is the bending moment factor expressed as:

$$ k = \frac{\cosh \Lambda}{\cosh \Lambda + 2\sqrt{2}\sinh \Lambda} \qquad (17) $$

with

$$ \Lambda = \frac{C}{2t_u} \sqrt{\frac{3F(1-v^2)}{2t_u E}} \qquad (18) $$

In the above equations, $b$ corresponds to the width of the bondline and $c/2$ to the distance from the middle of the bondline to the extremity of the bonded joint (half the overlap length).

With $v$ the Poisson coefficient of the adherent, $b$ the width, $c/2$ the distance from the middle of the bondline to the extremity of the bonded joint (half the overlap length).

For the overlap region, the authors obtained two equations, one for the adhesive shear stress (equation 19), and another one for peel stress distributions (equation 22).

### Adhesive shear stress distribution:

$$ \frac{\tau(x)}{\tau_s} = -\frac{1}{8} \left[ (1 + 3k) \frac{\delta}{\sinh(\delta)} \left( \cosh \frac{\delta x}{c} \right) + 3(1 - k) \right] \qquad (19) $$

$$ \text{With } \delta = \frac{c}{t_u} \sqrt{\frac{8G_a t_u}{E t_a}} \qquad (20) $$

$$ \text{With the average shear stress } \tau_s = \frac{2F}{c} \qquad (21) $$
Adhesive peel stress distribution:

$$ \frac{\sigma_y(x)}{Ft_u} \left(\frac{c}{2}\right)^2 = \frac{1}{\Delta} [A + B] \qquad (22) $$

$$ \text{With } A = \left( K_2 \lambda^2 \frac{k}{2} + \lambda k' \cosh(\lambda) \cos(\lambda) \right) \cosh\left(\lambda \frac{2x}{c}\right) \cos\left(\lambda \frac{2x}{c}\right) \qquad (23) $$

$$ \text{With } B = \left( K_1 \lambda^2 \frac{k}{2} + \lambda k' \sinh(\lambda) \sin(\lambda) \right) \sinh\left(\lambda \frac{2x}{c}\right) \sin\left(\lambda \frac{2x}{c}\right) \qquad (24) $$

$$ \text{With } K_2 = \sinh(\lambda)\cos(\lambda) - \cosh(\lambda)\sin(\lambda) \qquad (25) $$

$$ \text{With } K_1 = \cosh(\lambda)\sin(\lambda) + \sinh(\lambda)\cos(\lambda) \qquad (26) $$

$$ \text{With } \Delta = \frac{1}{2}(\sinh(2\lambda) + \sin(2\lambda)) \qquad (27) $$

$$ \text{With } \lambda = \frac{c}{2t_u} \sqrt[4]{\frac{6E_a t_u}{E_u t_a}} \qquad (28) $$

$$ \text{With } k' = \frac{v_0 c}{2Ft_u^3} = kF \sqrt{\frac{3(1-v^2)Ft_u}{E}} = \text{the transverse factor} \qquad (29) $$

Compared to the models that have been discussed so far, the Goland & Reissner's model is the first to consider an important phenomenon that can cause the failure of a bonded assembly: the peel stress. The obtained shear profile is closed to the one from Volkersen. The peel is in tension at the edge and compression in the central part. It is much more complete than the previous model derived from the Volkersen's approach but remains limited to linear elastic adhesives. As it can be seen in Figure 1.2-11, the Goland & Reissner model predicts higher adhesive shear stresses at the end of the overlap, mainly because of the peel stresses that cause additional shear stress [17].

<figure><img src="image_22.png" /><figcaption>Figure 1.2-11: Stresses predicted by Volkersen and Goland & Reissner models [22].</figcaption></figure>

The study of this model seems entirely appropriate in the case of application of bonded patches. However, a physical problem common to these analytical models (Volkersen and Goland and Reissner) is that they do not fulfill the zero-stress condition at the edges. This result can introduce error in the design of the ultimate capacity of bonded patches.

In addition, the limitation to linear elastic materials which may be not adapted for some structural adhesives.
1.2.1.4. Hart-smith approach

In 1973, Hart-Smith [18] proposed an improvement of the Volkersen's theory including the elasto-plastic (with perfect plasticity) behavior of the adhesive. As in Volkersen's model, the adhesive is supposed to remain in pure shear strain and the adherents in pure tensile strain. Historically, this development was carried out on symmetric double lap joints (Figure 1.2-12).

<figure><img src="image_23.png" /><figcaption>Figure 1.2-12: Application case of the Hart-Smith model.</figcaption></figure>

Based on these assumptions and equilibrium, two differential nonlinear equations can be obtained:

- one specific equation for the shear stress in the elastic zone, which is the same as the one from Volkersen,

- another one for the adhesive shear strain, $\gamma_p$, in the plastic zone of length $(c - d)/2$, $d$ being the elastic zone length.

As for the Volkersen's equation, the expression of the parameter $\omega$ remains similar (equation 30).

$$ \omega = \sqrt{\frac{G_a}{t_a}\left(\frac{1}{t_l E_l} + \frac{1}{t_u E_u}\right)} \qquad (30) $$

$$ \gamma(\xi) = \left(\frac{\omega^2}{2G_a}\right) \tau_p \xi^2 + C\xi + H \qquad (31) $$

$$ \xi = x - \frac{d}{2} \qquad (32) $$

$$ C = \frac{\omega \tau_p}{G_a} \tanh \left( \frac{\omega d}{2} \right) \qquad (33) $$

$$ H = \frac{\tau_p}{G_a} = \gamma_e \qquad (34) $$

$$ \gamma_e = \gamma \text{ at } x = \frac{d}{2} \qquad (35) $$

With $\gamma_e$ the adhesive elastic shear strain at yield, $\gamma_p$ the plastic part of the shear strain and $\tau_p$ the adhesive yield shear stress (Figure 1.2-13).

The strain continuity condition and the overall force equilibrium allow obtaining two equations that need to be solved to get access to the final shear stress state (equations 36 and 38).

$$ \left( \omega \frac{(c-d)}{2} \tanh \left( \frac{\omega d}{2} \right) \right)^2 = \tanh^2 \left( \frac{\omega d}{2} \right) + 2 \frac{\gamma_p}{\gamma_e} \qquad (36) $$

$$ \gamma_e + \gamma_p = \gamma \left( \xi = \frac{c-d}{2} \right). \qquad (37) $$

$$ \frac{\tau(x)}{\tau_p} = \left(1 - \frac{d}{c}\right) + \frac{\tanh\left(\frac{\omega d}{2}\right)}{\frac{\omega c}{2}} \qquad (38) $$
<figure><img src="image_24.png" /><figcaption>Figure 1.2-13: Perfect plastic law of the adhesive.</figcaption></figure>

A common shape of the typical solutions for the shear stress along the lap length is described in Figure 1.2-14.

<figure><img src="image_25.png" /><figcaption>Figure 1.2-14: Hart-Smith's shear stress distribution.</figcaption></figure>

The plateau corresponds to the plasticity around the edges of the bondline. The Hart-Smith model solved the main problem of the Volkersen and Goland & Reissner one, as it accounts for the plastic reduction of stresses around the free edges (Figure 1.2-15).

<figure><img src="image_26.png" /><figcaption>Figure 1.2-15: Relation between the overlap length and the adhesive shear stress distributions for Hart-Smith model, for a) short overlap and b) long overlap.</figcaption></figure>

Considering a longer bondline, the behavior is different as described on Figure 1.2-15a and b. For a low load level, the response is elastic within the bonded length. With the load increase, the yield stress is reached at one end first and a plastic zone starts to develop before the second end reaches the yield stress. Then, the plastic zone will develop on each side of the assembly. As the plastic zone continues to grow, the adhesive strain increases at the edges up to a maximal
allowable strain value and a crack initiate near the border of the overlap. If the load remains constant, the crack propagates and leads to failure. It can thus be observed that above a certain length, the ultimate strength and the overlap length are independent (Figure 1.2-16). Any bond defect in the center area (adhesion, material defect), where the stresses are equal to zero, should therefore have a limited impact on the overall strength of the patch [18].

<figure><img src="image_27.png" /><figcaption>Figure 1.2-16: Scheme presenting the relationship between the ultimate capacity and the bonded length.</figcaption></figure>

#### 1.2.1.5. Summary of the analytical models

The models described in the previous parts represent the main analytical approaches used for the study of mechanical states in bonded assemblies. Others have been developed with the aim of increasing the representativeness of analytical models for other types of applications or solicitations. The state of the art realized by Da Silva et al. in [17] and Gleich in [25] proposed a synthesis of those existing analytical models on the basis of the materials behavior (adhesive and adherents), the resolution method and the obtained physical quantities. From this review, it is important to note that in most cases there exists a specific length called the characteristic anchorage length that is necessary to transfer the stresses through the bonded joint. This parameter must be considered during the design. It is also important to be aware that in the case of structural adhesives, the adhesive rigidity effect cannot be neglected. In that case, shear and peel stress concentrations arise at the assembly edges.

Several limits exist to the application of those models, the main ones being the inadequacy with more complex geometries, especially at the edges [26], the hypotheses of linear behavior and the absence of peel stress computation. To overcome these issues, numerical approaches have been developed to determine the internal mechanical stress state either in the adhesive or at the interfaces.

### 1.2.2. Models based on numerical approaches

Before to present numerical approaches, the first paragraph will be dedicated to the presentation of the singularity difficulty associated with the analysis of such assembly. Then, three numerical approaches will be detailed presenting some examples of their applications for the study of adhesively bonded joint: the finite element approach, the discrete element approach and the M4 models.
1.2.2.1. Singularities

In the case of adhesive bonding, there is a numerical difficulty due to the mathematical singularities at the joint edges either due to the geometry or due to the presence of two materials with different rigidity. This depends on the edge treatments. Some commonly encountered ones are presented in Figure 1.2-17 indicating the presence of geometrical singularity (blue circles). Singularity refers to a point where, in linear elastic analysis, stress will reach an infinite value. This induces a high dependence of the obtained maximum stress values with mesh refinement.

<figure><img src="image_28.png" /><figcaption>Figure 1.2-17: Bonded joint possible singularities.</figcaption></figure>

To overcome this problem, adding a fillet of adhesive at the edge of the joint can help to reduce
its impact. Such fillet exists naturally in most of the realized joint [26]. It was shown that the
maximal stress value is dependent on the shape of this fillet. This effect was experimentally
studied by Zhao et al. [28] for a ductile and a brittle adhesive. The main formulated conclusions
are the following ones:

- joints bonded with a brittle adhesive are highly affected by the presence of a fillet and by its shape,

- joints bonded with a ductile adhesive present a better strength for sharp corners joints, when compared with round corner, indicating a lower sensibility to singularity.

1.2.2.2. Finite element method

Currently, the mostly used approach is the finite element method (FEM). It consists in a
numerical method using an iterative algorithm allowing to search for an approximate solution
of a partial differential equation U'' = f on a compact discretized domain with boundary
conditions. This method has been applied to adhesively bonded joints with success, depending
on the application case. L. Mouton in [28] showed accurate maximal strength prediction for a
single lap bonded tri-material (polyurethane/epoxy/steel) patch with FEM. You et al. [29] used
an elasto-plastic finite element analysis and experimental investigations to study the effects of
chamfer on the ultimate capacity of adhesively bonded steel single lap joints. The results
showed that all the maximum stresses in the bondline were reduced with the increase of the
chamfer angle. Additional review of the application of FEM on adhesively bonded joints have
also been made by Xiaocong et al. in [30].

To summarize, FEM is particularly adapted for the computation of stresses and strains for
complex geometries. There exist however some limitations in the case of adhesively bonded
joint application:

- the thickness ratio within the adherents and the center to edge length requires a fine mesh in these areas. Consequently, the number of degrees of freedom necessary to model a joint is generally high [31],

- the singularity locations, where the stress value increases with the mesh refinement, inhibits the convergence towards a solution. Therefore, this analysis should be used with care,

- those refined FEM tend to take a lot of time to compute, especially in the case of multiple supposed crack propagation path.
#### 1.2.2.3. Finite difference method

The finite difference method (FDM) is actually a way to solve a system of by approximating derivatives with finite differences. A series of equations, which can be numerically solved with matrix methods, are then created. It allows considering more complex boundary condition than with the closed-form solution.

This method implies determining the equations (an analytical model with assumptions) previously to the numerical calculations [32] [33].

This can be particularly adapted in the case of more complex geometries joint (tapered lap joint, scarf joint). The differential equations become more complex and the analytical solutions tend to become impossible to derive. The finite difference method (FDM) replaces the continuous variables in the differential equations by approximations. A series of equations, which can be numerically solved with matrix methods, are then created. It allows considering more complex boundary conditions than with the closed-form solution.

FDM has been used a lot before the massive development of commercial finite element software for the study of adhesively bonded joints. Thus, Bigwood et al. [34] compared the obtained stress profiles, in the case of a single lap joint, for both finite element and finite difference methods, and showed a good correlation.

To summarize, FDM seems to be highly relevant when a similar case is repeatedly encountered. In that case, it may limit computation time while accessing accurate results. It cannot however give access to very local information as FEM.

#### 1.2.2.4. M4 model

M4 models stands for Multiparticle Models of Multilayered Materials and was specifically developed to analyze edge stress fields at interfaces in the case of road pavement structures or composite layered structures [35]. It consists in simplifying the mechanical problems in layers and interfaces (Figure 1.2-18). Several versions of the M4 model exist depending on the kinematics hypotheses, the most famous one being the one developed by Pagano [36]. The construction of those models relies on four main steps:

- a polynomial approximation of the 3D stress fields in the thickness direction in each layer of the studied structure is carried out. This allows expressing forces, moments and shear internal forces for each layer, and shear and normal stresses at the interfaces,

- the associated generalized displacement fields are then approximated using the Hellinger-Reissner functional,

- the equations of equilibrium on the generalized internal forces and the boundary conditions for each layer allow obtaining the equations linking the internal forces with the interface stresses,

- finally, the constitutive equations are used to express the internal forces and interface stresses in function of the compliance of the materials.

Such an approach has been used in the case of adhesively bonded joints with elastic interface [37] or elasto-plastic interface [38]. Good results were obtained and a comparison with finite element approaches showed that, provided a refined mesh (in the width and length only, as there is no mesh in the thickness) was used and that the order of the M4 model was sufficient, a good correlation could be found.
<figure><img src="image_29.png" /><figcaption>Figure 1.2-18: Schemes of the M4 approaches.</figcaption></figure>

In conclusion on the numerical strategies review, it may be highlighted that, in the special case of bonded patch, special design with possible overlapping chamfer which allows optimized reduction of shear stresses at the interface and at the edge, the choice of the numerical strategy may be limited to the finite element method. However, cautions must be taken regarding the maximum stress values and their dependency with mesh size.

**1.3. Design approach and failure prediction**

There are few reliable failure criteria for bonded joints, and they generally depend on the application case. However, an accurate prediction of the strength of bonded joints is essential to reduce the number of costly tests at the design stage. Currently used approaches for predicting the strength of adhesively bonded joints are: the continuum mechanics approach (stress and strain based), fracture mechanics, and damage mechanics approach [31]. These approaches will be presented in more details in next paragraphs.

**1.3.1. Continuum mechanics approach**

Using the continuum mechanics hypothesis (material continuity), several authors got interested in critical stress or strain approaches. Of course, such approach is strongly related with the choice of the method used to determine those stresses or strains.

**1.3.1.1. Critical stress approach**

Initially, the critical stress approach was proposed for very brittle adhesive (linear elastic material) whose failure depends on the direction of maximum principal stresses [22]. With this approach, in the case of bonded joint, the maximal stresses are determined in the adherent, in the adhesive or close to the interfaces (depending on the observed failure mode) and then the values are compared to the maximal allowable ones.

Maximal shear stress has been widely used to predict ultimate joint strength (using closed-form analyses) under shear loading, considering the maximum allowable shear stress equal to the bulk adhesive shear strength. In a study led on single-lap joints, Da Silva et al. [39] have used the maximal shear stress determined with the Goland and Reissner model as a criterion. They showed that this criterion is only valid for brittle adhesives and for short overlaps. More recently, some designs based on such criteria have been done for the development of bonded
application, with low modulus thick adhesive and limited stress concentration, and showed good results [28]. Though the approach seems to be rather simple, some limitations may exist, especially the fact that the peel stress is ignored.

Mixed stress criteria have thus been developed to express the maximal allowable stress state considering multiaxial stress state.

Some studies have been made following standard continuous mechanics approach using the Von Mises yield criterion for an application case of GFRP scarf joint on steel [40]. The stress and strain distributions were obtained using elastic finite element analysis. It is important to note that the Von Mises stress is not dependent upon hydrostatic pressure part ($\Sigma_{i=1}^3 \sigma_{ii}/3$) of the stress tensor though this has a great influence on the adhesive's behavior [41].

The CIRIA Guideline (RP 645) and CNR Guideline (CNR_DT202) proposed therefore criteria expressions relying on the simplified Mohr's principal stress definition (Mohr circle), equations (39) and (40).

$$ \frac{\sigma_x + \sigma_y}{2} + \sqrt{\frac{(\sigma_x - \sigma_y)^2}{4} + \tau_{xy}^2} \le \frac{\sigma_p}{2} + \sqrt{\frac{\sigma_p^2}{4} + \tau_p^2} \quad (39) $$

$$ \frac{\sigma_y^2}{2} + \sqrt{\frac{\sigma_y^2}{4} + \tau_{xy}^2} \le \frac{F_{ak}}{\gamma_a \cdot \gamma_{rd}} \cdot \eta \quad (40) $$

With $\sigma_x$ being the loading direction stress, $\sigma_y$ being the normal direction stress, $\tau_{xy}$ the shear stress, $\tau_p$ being the adhesive yield shear stress and $\sigma_p$ being the adhesive normal yield stress, $F_{ak}$ the adhesive resistance evaluated through ISO standard single lap shear test, $\gamma_a$, $\gamma_{rd}$ two parameters ("partial factors") taking into account the conditions of the bonding (quality of the application and type of loading) and $\eta$ a compensating parameter depending of the type of loading and type of composite used.

In the particular case where $\sigma_y$ is negative, the overall value of the criteria will be superior to traction case. This expresses well that bonding under compression and shear sustains higher load than in traction and shear.

As already mentioned, for the finite element approach, the encountered singularity at the edges may prevent from obtaining mesh independent stress values at those locations. Alternative methods have also been proposed to use an effective stress at a certain distance of the edge to limit the impact of the singularity.

#### 1.3.1.2. Stress at distance

To overcome stress singularity, John et al. [42] used a critical distance combined with shear stresses to predict the maximal strength under linear elastic behavior. The main conclusion of these results is a global overestimation of the maximal strength of the bonding. This overestimation makes this criterion unusable in case of bonded patches with scarf design. Similarly, Crocombe [43] used a critical peel stress at a distance from the singularity with some success and concluded that the critical distance at which the criterion must be applied varies with the loading, since the type of loading will have an influence on the size of the plastic zone.
#### 1.3.1.3. Critical strain approach

For the study of more ductile adhesives, stress criteria are not appropriate anymore (failure is governed by strain). Hart-Smith [18] proposed the use of the maximum shear strain as a failure criterion. The main limitation of this approach is the necessity to determine the plastic strain. Another analysis has been proposed by Adams & Harris in 1984 [45], which studies the use of maximum principal strain in the adhesive for aluminum plate bonded with different epoxies in single lap joint. The main advantage of this criterion is the ability, following the direction of the maximal principal strain, to predict the failure mode.

#### 1.3.1.4. Critical plastic strain and energy density criteria

Critical plastic strain is a concept closed to the total strain energy criterion that states that failure (by yielding or by fracture) would occur when the total strain energy, or total strain energy density, reaches a critical value. The critical strain energy density $U_t$ is generally expressed in cartesian coordinates by equation (41) [45].

$$U_t = \frac{1}{2}(\sigma_{xx}\varepsilon_{xx} + \sigma_{yy}\varepsilon_{yy} + \sigma_{zz}\varepsilon_{zz} + \tau_{xy}\gamma_{xy} + \tau_{xz}\tau_{xz} + \tau_{yz}\tau_{yz}) \quad (41)$$

The critical plastic strain is restricting the analysis only to the plastic part of the strain ($\varepsilon_{total} = \varepsilon_{elastic} + \varepsilon_{plastic}$). This approach has been used by Adams and Harris [26] as a failure criterion. This concept is closed to the critical plastic length proposed in [41]. Zhao et al. [27] also applied this criterion combined with a distance from the free edge to express a maximal allowable energy plastic density before the failure of the assembly. The advantage of the use of an energy density is the limited sensitivity to the mesh size. It is common knowledge that the accuracy of the Finite Element (FE) approach is more reliable when it is interpreted as an average, rather than in a local sense [22]. Using this criterion, Zhao et al. [27] used the adhesive thickness as an integration zone for the energy density computation of ductile adhesives. The main issue with this approach is an experimental issue, and is related to the difficulties to measure the plastic strain in samples especially for low thicknesses.

All the above criteria are applicable to continuous structures. When discontinuity such as crack occurs, it becomes necessary to define new criteria based on fracture mechanic approach.

### 1.3.2. Fracture mechanics

The Linear Elastic Fracture Mechanics (LEFM) is interested in the study of cracks in linear elastic isotropic materials. It assumes that its propagation is associated with 3 main modes: mode I that corresponds to opening effect, mode II that corresponds to in-plane shear effect, and mode III that corresponds to out-of-plane shear (Figure 1.3-1).

<figure><img src="image_30.png" /><figcaption>Figure 1.3-1: The three modes of loading of a crack.</figcaption></figure>
The LEFM is valid when inelastic strains are small compared to the size of the crack. If large plastic strains are expected, the Non-Linear Fracture Mechanics (NLFM) might be used [46]. Two main approaches can be detailed in the LEFM:

- the direct method: the stress state around the crack tip is determined using the stress intensity factor Ki (i = I, II, III regarding the loading mode of the crack). The stresses are defined, depending on the geometry and loading. In the case of bi-material joint, Golio et al. [47] showed the difficulties to define the amplitude of the singular field around the crack tip. Even more difficulties can be expected in the case of multiple material bonded joint with thin bondline,

- the energetical approach: this approach is based on the Griffith's theory and will be discussed in the following paragraph.

#### 1.3.2.1. Griffith's theory

The fracture mechanics approach proposes to define the crack propagation by quantifying the energy released by an increment of crack length $da$ (Figure 1.3-2). In 1920, Griffith [48] proposed an energy balance to explain crack propagation in the case of a linear elastic material under static load. In the vicinity of an equilibrium position, to allow the propagation of the crack, the energy released for an increment of crack extension ($b*da$), $G(a)$, is equal to the variation of stored elastic energy ($U$) (red and blue triangles in Figure 1.3-2 represent the unloaded area before and after crack propagation) and the work of the boundary/external loads ($F$) for an increment of crack length (see equation (42)). The difference between the stored elastic energy and the work of the boundary load is noted $\Pi = U - F$.

$$G(a) = -\frac{d(U-F)}{b \, da} \qquad (42)$$

<figure><img src="image_31.png" /><figcaption>Figure 1.3-2: Energy released with the crack propagation.</figcaption></figure>

Two cases can then be described for linear elasticity (Figure 1.3-3).

- In the case of a prescribed load, with $\Delta$ the general displacement at the loaded point:

$$F = P\Delta, U = \frac{P\Delta}{2} \rightarrow G(a) = -\frac{1}{b}\left(\frac{d\Pi}{da}\right)_P = \frac{P}{2b}\left(\frac{d\Delta}{da}\right)_P \qquad (43)$$
The crack extension $da$ results in a net increase in strain energy.

- For a fixed displacement:

$$F = 0, U = \frac{P\Delta}{2} \rightarrow G(a) = -\frac{1}{b}\left(\frac{dU}{da}\right)_{\Delta} = -\frac{\Delta}{2b}\left(\frac{dP}{da}\right)_{\Delta} \quad (44)$$

<figure><img src="image_32.png" /><figcaption>Figure 1.3-3: a) Cracked plate at initial state, b) Cracked plate after fixed displacement $\Delta$ or fixed loading $P$.</figcaption></figure>

Thus, $U - F = -\frac{P\Delta}{2}$ for the fixed load case, or $U - F = \frac{P\Delta}{2}$ in fixed displacement case.

The energy release rate (ERR) can then be calculated using equation (45).

$$G(a) = \frac{1}{2b} \frac{d(\pm P(a)\Delta(a))}{da} = \frac{P^2}{2b} \frac{dC}{da} \quad (\text{Irwin-Kies equation [45]}) \quad (45)$$

With $C$ the compliance ($C=\Delta/P$).

In case of nonlinear materials, if the non-linearity is confined to a small region near the crack tip (Small Scale Yielding assumption), the LEFM may be applied [41].

In conclusion, the propagation of an internal defect (such as crack) will occur when the available energy at the defect tip ($G$) equals the energy needed for the crack to propagate ($G_c$, the critical energy release rate, or critical toughness). The value $G_c$ is directly linked to the material, type of failure (cohesive or adhesive) and the mode of loading. Many energetic failure criteria can be found in the literature for the combination of modes. Yet, using simple elastic hypotheses, it may be possible to have access to the value of $G_c$ using the applied load and resulting displacement. However, the measurement of the crack propagation is, in most cases, a pre-requisite. This approach has one main limitation. To be used, a pre-crack must be specified to allow computing the energy release rate when the crack propagates from $a$ to $a + da$.
#### 1.3.2.2. J-integral

In 1967, Cherepanov [50] and Rice [51] developed a new method based on the determination of an energy flow through a surface. The concept of the J-integral is based on the independency of an energetic contour path integral with the path located around the crack tips for homogeneous elastic materials. The direct mathematical expression of the J-integral in 2D is given in equation (46). The coordinate system as well as an example of a contour path can be seen in Figure 1.3-4. The first term in the integral represents the elastic strain energy while the second term represents the work done by external loads on the outline.

<figure><img src="image_33.png" /><figcaption>Figure 1.3-4: J-integral contour path Γ around a crack tip in two dimensions.</figcaption></figure>

$$J = \int_{\Gamma} w(\varepsilon) dy - \sigma_{ij} \cdot n \frac{du_i}{dx} d\Gamma \quad (46)$$

With $w(\varepsilon)$, the strain energy density, $\sigma_{ij}$, the stress tensor, $u_i$ the displacement vector, $\Gamma$ the contour path and $n$ the vector normal to the contour path.

The following main properties must be mentioned:

- the J-integral is independent of the contour path (as long as the contour embeds the crack tip),

- the path goes in the counter-clockwise direction from the lower crack surface to the upper crack surface,

- the crack must not be loaded.

The J-integral represents the energy release rate if the crack grows straight ahead (for planar crack grows) [52] [53]. In the case of elasto-plastic materials, Rice showed that if monotonic loading is assumed (with no crack propagation) then the J-integral can also be used to compute the critical energy release rate ($J_c = G_c$). Some good results have been found with the use of the J-integral method. Choupani [54] used it to determine the critical toughness $G_c$ with J-integral approach in the case of thick adhesive and adhesive failure with Arcan test.

#### 1.3.2.3. Mixed mode criteria

The physical quantities used to characterize the bonded joints often refer to the critical energy release rate ($G_c$) which has been abundantly used to characterize and predict failure of bonded joints [55], [56].

In design, some references consider only the mode I (DNV ST0376) indicating it is the worst-case situation and the most common type of failure.

For the studied application, the loading is mainly in shear. Therefore, the study of the fracture in only mode I should not be sufficient, and special attention should be provided to the effect of mode II. The necessity to define a fracture criterion that correctly establishes the fracture
toughness ($G_c$) in mixed mode is related to the incapacity to test every mixed-mode combination. Several mixed mode failure criteria have been used in the literature [54].

In the case of adhesively bonded joint, the most commonly used criteria are the Power law criterion [58] (equation 47) and the B-K criterion [59] (equation 48).

$$ \text{The power law criterion: } \left(\frac{G_I(a)}{G_{Ic}}\right)^\alpha + \left(\frac{G_{II}(a)}{G_{IIc}}\right)^\beta \geq 1 \qquad (47) $$

$$ \text{The B-K criterion: } G(a) \geq G_{Ic} + (G_{IIc} - G_{Ic}) \left(\frac{G_{II}(a)}{G(a)}\right)^\eta \qquad (48) $$

where $G_{Ic}$ is the critical strain energy release rate in pure mode I, $G_{IIc}$ is the critical strain energy release rate in pure mode II and $\alpha, \beta, \eta$ being values that are numerically fitted to the experimental data.

### 1.3.3. Coupled stress-energy method

The principle of the coupled criteria approach is to compensate the major problems of the main criteria of continuous and fracture mechanics applied to bonded area.

For continuous mechanics, the critical stress approach is relevant to express the mechanical state of materials following the conditions of the continuous mechanics. However, it is generally inefficient to express this state around geometrical singularity. The definition of a criteria following these values will systematically lead to an error on crack initiation prediction.

For fracture mechanics, the definition of the critical value, that allows the propagation of a crack through the materials, cannot be applied to an uncracked assembly (fracture mechanics theory assumption). As defined by Leguillon [60], these two criteria are expressing a paradox, as the first one is nearly always fulfilled because of the stress concentration and the other cannot.

Moreover, generally none of those criteria matches the results obtained from the experiment.

The experiments of Parvizi et al. [61] followed by the analysis of Leguillon [60] proposed an explanation to this paradox. They defined a special traction test that correlated the crack initiation with the applied load and the stress state. They have proved that, while the crack develops, the stress criterion is not relevant anymore to describe the crack propagation. They also proved that, when the crack is opened, the energy criterion governs the process and the stress criterion remains always fulfilled.

The main conclusions of their work are that a crack initiation will occur when those criteria are fulfilled simultaneously, and both are needed to express a predictive global criterion. The method is detailed step by step in the following parts.

#### 1.3.3.1. Stress criteria

The direct application of part 1.3.1.1 allows defining different stress criteria, depending on the type of loading. The stress criterion is here expressed as a function $f$ depending on the set-up and the level of load. It is expressed all along the bonded joint for each $x$ positions.

For mode II:

$$ f(x) = \frac{\tau_p}{\tau(x)} \qquad (49) $$

For mode I:

$$ f(x) = \frac{\sigma_p}{\sigma(x)}, \quad \sigma(x) > 0 \qquad (50) $$
$\tau_p$ being the adhesive yield shear stress and $\sigma_p$ being the adhesive yield stress.

Criteria fulfillment corresponds to the fact that the stress at location (x) is equal or greater than the allowable stress, leading to f(x) being inferior or equal to 1.

For mixed mode I and II, similar approach as the one detailed in part 1.3.1 can be defined for stresses application. For example, a power law type criterion depending on shear and peal stress can be used, equation (51).

$$f(x) = \frac{1}{\sqrt{\left(\frac{\tau(x)}{\tau_p}\right)^2 + \left(\frac{\sigma(x)}{\sigma_p}\right)^2}} < 1 \qquad (51)$$

Other criteria could also be used for the mixed mode, as the Von Mises yield criteria or the CIRIA criteria, described in part 1.3.1.1.

<figure><img src="image_34.png" /><figcaption>Figure 1.3-5: Typical analytical stress state and stress criteria curve.</figcaption></figure>

On Figure 1.3-5, it can be seen that for a defined level of loading, all the area within $x = ]0; a_d[$ experience stresses above the allowable limit. In this area the stress criterion is smaller than 1 corresponding to the damaged zone.

### 1.3.3.2. Energy criteria

The same kind of criteria can also be applied to the energetic analysis defined by a function h.

For mode I:

$$h(x) = \sqrt{\frac{G_{Ic}}{G_I(x)}} < 1 \qquad (52)$$

$G_I(x)$ being the computation of the strain energy release rate in mode I.

For mode II:

$$h(x) = \sqrt{\frac{G_{IIc}}{G_{II}(x)}} < 1 \qquad (53)$$
$G_{II}(x)$ being the computation of the strain energy release rate in mode II.

For mixed mode I and II, formulations detailed in previous part can be used, as for example, the Power law type criteria (equation (54)). Figure 1.3-6 presents an example of mixed mode criterion based on a power law. It can be seen that for a defined level of loading, all the area within $x = ]0; a_c[$ has a lower energy level than the one required to have a crack propagation (the criterion is smaller than 1). Once the crack length is superior to this critical length, at this load level, the crack propagates.

$$h(x) = \frac{1}{\sqrt{\left(\frac{G_I(x)}{G_{Ic}}\right)^\alpha + \left(\frac{G_{II}(x)}{G_{IIc}}\right)^\alpha}} = 1 \qquad (54)$$

<figure><img src="image_35.png" /><figcaption>Figure 1.3-6: Energy release rate and energy criteria along the bonded length.</figcaption></figure>

The use of this criterion allows defining a minimal allowable crack length before the crack propagates. Numerical modeling can be used to determine this energy for different crack lengths through the Crack Closure Technique (CCT) or Virtual Crack Closure Technique (VCCT).

In the crack closure method or technique, the crack is physically extended, or closed, during two complete finite element numerical steps. This approach is based on Irwin's crack closure integral [60] and on the assumption that the energy released $d\Pi$ when a crack is extended by a length $da$ is identical to the energy required to close the crack around the crack tip. In the first step with a crack size $a$, the constraint forces at the crack tip will be retrieved. In the second step, the node at the crack tip will be separated and thus the crack size in the second model will be $a + da$. In the case of the VCCT, this step is automatically done. The displacements of the just separated nodes will then be retrieved in each direction, allowing to compute the energy release rate in each mode (Figure 1.3-7). In the case of the VCCT method, the determination of the internal forces necessary to keep the crack closed is carried out at the end of the second stage when the crack has propagated by the length $da$.
<figure><img src="image_36.png" /><figcaption>Figure 1.3-7: Two-steps VCCT overview.</figcaption></figure>

Once these forces and displacements have been retrieved from the model, the total strain energy release rate as well as the contributions from mode I and mode II can be obtained by using the equation (55) [63].

$$G = \frac{1}{2\Delta_a} [F_{c,x}(u_c - u_d) + F_{c,y}(v_c - v_d)] = G_I + G_{II} \quad (55)$$

The first term corresponds to the mode II contribution and the second one corresponds to the mode I contribution.

#### 1.3.3.3. Stress-energy criterion

For a given load level, the use of a coupled stress-energy criterion allows comparing the length of the damaged bonded joint, $a_d$ (stress criteria), to the minimum crack length necessary for crack propagation, $a_c$ (energy criteria). When the load is such that the crack length $a_c$ equals the damaged length $a_d$, it corresponds to the critical load leading to failure (Figure 1.3-8).

<figure><img src="image_37.png" /><figcaption>Figure 1.3-8: Coupled stress-energy criterion at critical load level (the extent of the damaged zone is equal to the minimum propagation of the crack).</figcaption></figure>

The coupled approach requires several parameters to be characterized ($\tau_p$, $\sigma_p$, $G_{Ic}$, $G_{IIc}$). It also requires the calculation of the energy release rate curve and the stress curve to completely define the criterion. However, it remains interesting as it both considers the uncracked structure state and its ability to sustain damages and cracks.
**1.4. Damage mechanics and cohesive zone modeling**

Some authors have proposed another approach that relies on the definition of an interfacial
behavior and that combines both fracture mechanics and continuum mechanics failure criteria
to overcome the main encountered difficulties and limits. It is called the Cohesive Zone
Modeling (CZM). The CZM allows focusing only on the interface behavior (independently of
the rest of the model) in order to simulate more precisely the behavior of the bonded joint and
thus its resistance.

**1.4.1. Description and concept**

Historically, the first definition of a cohesive zone model has been proposed by Barenblatt [64], in 1962 and Dugdale [65] in 1960. This description was made to try to solve the problem of stress singularity around a crack tip in the case of brittle fracture mechanics to model the separation of atomic bonds at the tip of the crack.

These models assume that the damaged area in front of the crack (fracture process zone) can be
described as an interface where the displacement field can admit discontinuities, while
continuing to transmit load. This cohesive process area constitutes a transition zone between
the undamaged material and the broken material where the damage can grow. Figure 1.4-1
shows the described zone and the resulting interfacial displacement.

<figure><img src="image_38.png" /><figcaption>Figure 1.4-1: Representation of the fracture process zone and a bilinear cohesive zone law for mode I.</figcaption></figure>
The cohesive zone model represents the cohesive stress versus the interface's relative displacement evolution. This corresponds to the traction separation law in mode I. The initiation of the damage is connected to the maximum values of the different stresses at the interface and to the creation of the process zone in the bondline. Damage then grows with interface's relative displacement. When the stored energy in the cohesive element is equal to the critical energy release rate of the material, the traction or shear forces are reduced to zero, and a crack front is created.

Hillerborg [66] introduced in 1976 a bilinear law with three main parameters expressing the interface law:

- $\sigma_c$ the maximal material stress that the material can sustain,

- $G_{ic}$ the critical energy release rate, which is the energy needed for the separation of the interface,

- $\delta_n$, the ultimate slip that the material can sustain before separation (for respectively peel and shear law).

Numerically, CZM laws are established between nodes of cohesive elements. Figure 1.4-2 presents the local displacement allowed by the cohesive law for each node at the interface (cohesive element) depending on the loading mode; $\delta_n$ for local opening (mode I), and $\delta_t$ for local sliding (mode II).

<figure><img src="image_39.png" /><figcaption>Figure 1.4-2: Four-node linear cohesive elements separation in FEM.</figcaption></figure>

There are two options to define cohesive law between two materials:

- use the cohesive elements (with defined thickness) to represent the adhesive layer (Figure 1.4-3a),

- or represent the adhesive layer with volume elements and use interfacial cohesive element (with zero thickness) to model the interface's behavior (Figure 1.4-3b).
<figure><img src="image_40.png" /><figcaption>Figure 1.4-3: Models: a) with thick cohesive zone model representing the adhesive layer between the adherents, b) with zero thickness cohesive elements with adhesive volume elements.</figcaption></figure>

Using the first methodology (cohesive elements with defined thickness), the elastic loading and the plastic dissipations of the adhesive bond are simulated by solid finite elements. In the case of highly ductile, or thick adhesive with cohesive failure, the introduction of 3D elasto-plastic element should be really efficient.

In this case, the stiffness of the interfacial cohesive elements can be defined as follows:

- stiffness in tension is given by (equation (56)):

$$k_I = \frac{ES}{t} \qquad (56)$$

- shear stiffness is given by (equation (57)):

$$k_{II} = \frac{GS}{t} \qquad (57)$$

With E, G being respectively the Young modulus and shear modulus of the material, S the surface of the element (distance between nodes 1 and 2 multiplied by width in Figure 1.4-2), and t its thickness. It is to be noted that these values are generally obtained through bulk test. However, in the case of thin bonded joint, tests such as Arcan test for multi axial properties, or TAST test for shear properties seem to be more appropriate to express the particular materials properties as described in [67].

A few works on CZM technique use the zero-thickness approach to model the bonded joint behaviour [68], [69]. This approach should be relevant in the case of thick bonded layer or multiple interfaces composites failure. In the other hand, Sorensen [70] and Yang [71] used the thick cohesive elements to accurately express the large plastic strain impact around the crack tip of ductile adhesive and to determine the resulting cohesive law without having to obtained directly $G_c$.
1.4.2. Cohesive law shape

The cohesive law shape contains information from underlying physical processes at the microscale. Thus, the stiffness of the cohesive elements is inversely proportional to their thicknesses. Therefore, the definition of the cohesive law will change, depending on the type of cohesive element. The first initial slope will be infinite for zero thickness cohesive element (Figure 1.4-4, Figure 1.4-5), and should be linked to adhesive's Young modulus, or shear modulus (depending on the mode) for thick cohesive element (Figure 1.4-6, Figure 1.4-7, Figure 1.4-8).

Its determination is important for accurate mechanical modeling. This is especially true when the fracture process zone is large [72]. There exist cohesive zone models with many different shapes for the cohesive law. The following examples are mostly dedicated to mode I. Yet, similar laws have been used for mode II.

The first introduction by Dugdale [65] of a cohesive law in 1960 followed a pure plasticity scheme (Figure 1.4-4). This model describes the relation between the tensile force and the normal displacement jump. This jump is null as long as the stress does not reach the critical value $\sigma_c$. Then, when the displacement or slip exceeds the threshold value, interaction between the lips is cancelled and thus no stress can be transmitted.

<figure><img src="image_41.png" /><figcaption>Figure 1.4-4: Dugdale cohesive law.</figcaption></figure>

Needleman [73] suggested the introduction of cubic and polynomial laws to express accurately the failure of materials following nonlinear properties (Figure 1.4-5).

<figure><img src="image_42.png" /><figcaption>Figure 1.4-5: Needleman cohesive law.</figcaption></figure>

Tvergaard [74] used a trapezoidal form to model the behavior of an elasto-plastic adhesive joint (Figure 1.4-6).
<figure><img src="image_43.png" /><figcaption>Figure 1.4-6: Trapezoidal cohesive law.</figcaption></figure>

Camacho [75] (Figure 1.4-7) used a bilinear tensile-separation relationship applied to delamination problems in layered composite [76], [69]. This model is also commonly used to simulate the strength of bonded composite joints as it is accurate to express failure without plasticity, often encountered in adhesive failures [68].

<figure><img src="image_44.png" /><figcaption>Figure 1.4-7: Bilinear cohesive law.</figcaption></figure>

Petersson [77] defined a softened "bilinear" law which is frequently used to model concrete or wood failure because of its high accuracy to express delamination or bridging (in this case, some fibers remain attached to each adherent after the crack has already propagated) (Figure 1.4-8). This law needs at least one more parameter than the bilinear cohesive law, increasing the difficulty of its definition. However, the softened bilinear behavior seems to be a good compromise between the polynomial law and the bilinear one.

<figure><img src="image_45.png" /><figcaption>Figure 1.4-8: Peterson cohesive law.</figcaption></figure>

The exact shape of the cohesive law can nowadays be extracted from experimental tests [78], although the tests are not standardized. If the cohesive laws have been obtained experimentally,
the available cohesive law shape and parameters of the model should be selected to best fit the experimental data. If the shape of the cohesive law is not known and there is no previous experience on the most appropriate selection, the simplest model is to assume linear softening. Additionally, the cohesive law shape may have an impact on the computational performance depending on the type of studied material [79].

For mixed mode cohesive law, several expressions have been proposed to express the impact of mode mixity on the interface properties. The simplest one is the power law expressed by equation (58).

$$1 = \frac{G_I}{G_{Ic}}^{\alpha} + \frac{G_{II}}{G_{IIc}}^{\beta} \qquad (58)$$

Where the exponents $\alpha$ and $\beta$ are determined by a curve fit to the fracture toughness data. For $\alpha=\beta=1$, this criterion becomes the linear interaction criterion, where the $G_I$ component of the experimental data is simply plotted versus its $G_{II}$ component.

Other criteria have been discussed in detail in several overview studies [80]. Today, the most often used 2D fracture criterion is the so-called B-K criterion proposed by Benzeggagh and Kenane [81] where a simple mathematical relationship between $G_c$ and $G_{II}/G_T$ is used in equation (59).

$$G_c = G_{Ic} + G_{IIc} - G_{Ic} \left( \frac{G_{II}}{G_c} \right)^{\eta} \qquad (59)$$

With $G_T$ the energy release rate in mixed mode. The exponent $\eta$ is determined by a curve fit to the fracture toughness, $G_c$, plotted as a function of the mixed-mode ratio, $G_{II}/G_c$. The parameters $G_{Ic}$, $G_{IIc}$, and $\eta$ are required inputs to perform modeling.

The surface of the cohesive law (depending of mixed mode ratio) can thus be obtained following a given criterion (power law, B-K) or a polynomial surface approximation as visible in Figure 1.4-9 [82].
<figure><img src="image_46.png" /><figcaption>Figure 1.4-9: Mixed mode cohesive law surface (Power law criterion, [83]).</figcaption></figure>

Several studies [84] [76] have been carried out on the expression of critical toughness in mixed mode, and it has been shown that the criterion B-K [81] and Power law could succeed in expressing $G_c$ with composite (UD and angle-ply) test samples. More particularly, Camanho et al. [75] showed, for UD composites with different types of resin, that, with the correct choice of $\eta$ for the B-K criterion, the critical toughness (for large mixed mode ratio) can be represented. A comparison between the Power law and the B-K criterion has also been done in [85] concluding that the Power law criterion, which is one of the most popular criteria, is not able to capture the rising mode I component in the low mode II ratio region for mixed mode test. The Power law criterion is also difficult to express in terms of the mixed-mode ratio ($G_{II}/G$). The B-K is then often used to produce criteria in accordance with experimental results. It has the advantage to use only 3 curve fitting parameters ($G_{Ic}$, $G_{IIc}$ and $\eta$) (cf. equation (59)).

### 1.4.3. Determination of the critical energy release rate in mode I according to standardized tests

The double cantilever beam (DCB) test is the most commonly used method for measuring the initiation and propagation values of Mode I fracture energy $G_{Ic}$ under static and cyclic loading conditions (see ASTM D5528 [87], ASTM D3433 [86], [88]) (Figure 1.4-10). A tensile load is applied to a DCB specimen with an embedded through-width insert at the specimen mid-plane which creates the initial crack of length $a$. The tensile force is applied perpendicular to the crack surface $P$ and the local opening at the load application $\delta$ is monitored. Crack length is measured using either a travelling microscope or video camera. Building and testing of DCB specimens is straightforward and relatively inexpensive. Figure 1.4-11 presents a typical DCB test $P(\delta)$ curve which can be described in two parts. The first one is the elastic loading of the adherent in bending, which may finish with a non linear part because of the creation of the process zone near the crack tip. The second part, where the load decreases because of the crack propagation, leads to an increase of the length of the adherent in bending.
<figure><img src="image_47.png" /><figcaption>Figure 1.4-10: DCB test ([87]).</figcaption></figure>

<figure><img src="image_48.png" /><figcaption>Figure 1.4-11: Exemple of DCB P(δ) curve.</figcaption></figure>

Different methods were proposed to determine the energy release rate from such a test. Four of them are described here and are from two different standards: the modified beam theory method, the compliance calibration method, the modified compliance calibration method, and the shear corrected beam theory method.

- the Modified Beam Theory Method (MBTM):

This method, detailed in ASTM D5528 [87](for composite-to-composite delamination) follows the beam theory which tends to overestimate the critical toughness. In that case, the expression of the energy release rate is then described by equation (60).

$$G_I(a) = \frac{3P\delta}{2b(a + |\Delta|)} \qquad (60)$$

With $\Delta$ a compensation factor (which is equal to 0 in the “normal” beam theory method) for the rotation of the adherent upstream of the crack front. It must be experimentally determined by generating a least square plot of the cube root of compliance ($C^{1/3}$), as a function of the crack length as shown in Figure 1.4-12.

<figure><img src="image_49.png" /><figcaption>Figure 1.4-12: Typical DCB C(a³) curve.</figcaption></figure>
- Compliance Calibration Method (CCM):

This second method, detailed in ASTM D5528 [87], gives the expression of the energy release rate in the equation (61).

$$G_I(a) = \frac{n\delta P}{2ba} \qquad (61)$$

with n the slope of least square regression of the equation $log(C) = f(log(a))$.

- Modified Compliance Calibration Method (MCCM):

The last method detailed in ASTM D5528 [87] proposes another computation of $G_I(a)$ given in equation (62).

$$G_I(a) = \frac{3P^2 C^{\frac{2}{3}}}{2A_1bh} \qquad (62)$$

With $A_1$ the slope of the curve: $\frac{a}{h} = f(C^{\frac{1}{3}})$

- Shear Corrected Beam Theory Method (SCBTM):

This method is detailed in the ASTM D3433 [86] (for steel-to-steel bonding), and is derived from the LEFM and the beam theory. It proposes another expression of the relation between the variations of the compliance versus the crack length (equation (64)).

$$\frac{dC}{da} = \frac{8}{Eb} \left( \frac{3a^2}{t_u^3} + \frac{1}{t_u} \right) \qquad (63)$$

$$G_I(a) = \frac{4P^2}{Eb^2} \left( \frac{3a^2}{t_u^3} + \frac{1}{t_u} \right) \qquad (64)$$

E being the Young modulus of the adherents and $t_u$ the thickness of the adherent.

Several initiation $G_c$ values may be determined from the load/displacement curve as described by the ASTM standards. These values are based on the energy stored at the beginning of the non-linearity of the load/displacement curve, or at the maximal load (beginning of the crack propagation). However, they are highly dependent on the quality of the sample crack initiation process and tend to be lower (more conservative) than the material (or interface) crack propagation critical toughness. This value is obtained when $G_{Ic}$ becomes independent on the crack length, therefore, when the critical toughness curve $G(a)$ becomes stable as visible in Figure 1.4-13.
<figure><img src="image_50.png" /><figcaption>Figure 1.4-13: G(a) curve.</figcaption></figure>

### 1.4.4. Determination of the critical energy release rate in mode II according to standardized tests

The only standardized experimental setup for the mode II critical energy release rate is the End Notch Flexure (ENF), (ASTM D7905 [89], Figure 1.4-14). It is essentially a DCB specimen loaded under three-point flexure. The dimensions are similar to those employed for the DCB specimen.

<figure><img src="image_51.png" /><figcaption>Figure 1.4-14: ENF sample.</figcaption></figure>

The two methods previously exposed for mode I (Compliance Calibration Method CCM and modified beam theory method MBTM) can also be applied for mode II. The CCM method has been proved to be accurate and is recommended in the ASTM D7905 standard [90]. The alternative method, the MBTM, tends to underestimate $G_{IIC}$ (depending on material properties) because of the absence of corrections to account for shear strain [14].

The following experimental issues with the ENF test have been listed:

* for brittle adhesives, or adhesive failure, uncontrolled crack propagation can be expected [14] [91],

* in [90], it has been shown that the ENF test does not guarantee the observation of pure mode II over its entire length. Indeed, there exists mixed mode that can decrease the obtained value of the critical strain energy release rate,

* too little crack propagation length might not be enough to observe a stabilized $G(a)$ curve into $G_c$, [14].

The ASTM standard (ASTM D7905 [89]), for composite-to-composite bonding, proposes simplifying hypothesis for the CCM approach as unstable crack propagation may occur during the ENF test. The same sample is used three times with different pre-cracked lengths $a_0$.
After the preliminary investigations, for each $a_0$, the three-point bending test is then performed until the crack propagates. The main hypothesis of this test is that the crack propagates when the maximum force ($P_{max}$) is reached, as visible in Figure 1.4-15 and $G_{IIc}$ can therefore be computed from the equation (65).

<figure><img src="image_52.png" /><figcaption>Figure 1.4-15: Exemple of ENF P(δ) curve.</figcaption></figure>

$$G_{IIc} = \frac{3 m P_{max}^2 a_0^2}{2 b} \qquad (65)$$

$$C = A + m a^3 \qquad (66)$$

With $m$ (slope) and $A$ (intercept), being the coefficients of the linear least square regression analysis of the compliance curve determined by fitting (equation 66).

It is to be noted that the compliance curve must be obtained using a minimum of three points (three tests with one sample). The main interest of this approach is that $G_{IIc}$ is directly obtained using equation (65), without precise measurement of the crack during the whole test.

### 1.4.5. Determination of the critical energy release rate in mixed mode according to standardized tests

The most frequently used experimental setup is the Mixed Mode Bending test (MMB) (Figure 1.4-16) which is described in standard ASTM D6671 [101]. It is based on a mix of DCB and ENF tests and allows testing under the full range of mixed mode ratio, from pure Mode I to pure Mode II. In MMB test, a lever is used to load the specimen. Loads are applied at the center of the specimen which creates mode II loading and at the upper bonded tabs position which creates mode I loading. The mixed mode ratio can be easily controlled through the displacement on the lever arm and the change of the length $c$ (Figure 1.4-17).

<figure><img src="image_53.png" /><figcaption>Figure 1.4-16: MMB test bed and sample.</figcaption></figure>
<figure><img src="image_54.png" /><figcaption>Figure 1.4-17: MMB Test configuration.</figcaption></figure>

In the ASTM D6671 standard, the analytical solutions for the critical toughness computation are obtained using Corrected Beam Theory (CBT) proposed by Kinloch et al. [89]. In [90], Reeder proposed, based on the CBT, for only mode I and mode II loading conditions, the following equations to evaluate the critical toughness value for isotropic adherents, monitored during the crack propagation (blue curve in Figure 1.4-18):

$$GI = \frac{12P^2(3c - L)^2}{16b^2h^3L^2E} (a + \chi h)^2 \quad (67)$$

$$GII = \frac{9P^2(c + L)^2}{16b^2h^3L^2E} (a + 0.42\chi h)^2 \quad (68)$$

These equations connect the toughness to the lever length c, the loading P and the crack length a of the sample, the Young's modulus of the adherents E, and the shear modulus of the adherents G, the length of the sample L, its thickness and width, h and b.

$$\chi = \sqrt{\frac{E}{11G}\left\{3 - 2\left(\frac{\Gamma}{\Gamma + 1}\right)^2\right\}} \quad (69)$$

$$\Gamma = 1.18 \frac{E}{G} \quad (70)$$

$$G_{tot} = GI + GII \Rightarrow Gc \quad (71)$$

Where $\chi$ is the crack length correction parameter and $\Gamma$ is the transverse modulus correction parameter.

Figure 1.4-18 shows typical load/displacement (P/δ) curves. The monitoring of these values must allow computing the toughness curve depending on the mode ratio. However, it is specified that in case of uncontrolled crack propagation (generally at high $G_{II}/Gc$ ratio, see green curve on Figure 1.4-18), an approximation of $G_{Ic}$ and $G_{IIc}$ (initiation critical toughnesses) can be made using the critical toughness computation at crack initiation, using the maximal load $P_{max}$ and the initial crack length $a_0$. Nonetheless, it may cause an underestimation of $Gc$ value.
<figure><img src="image_55.png" /><figcaption>Figure 1.4-18: MMB exemple P(δ) curve.</figcaption></figure>

### 1.4.6. Non-standard test Gc determination

#### 1.4.6.1. In Mode I

The adhesion between thin surface and a flat metal surface is commonly made through the blister test. The setup is based on the formation of a blister by injecting gas under pressure between a film and a substrate as shown in Figure 1.4-19. The mode I critical toughness is determined by recording the pressure as a function of the variation of the blister radius. The blister shape is monitored in 2D to measure the radius at different increasing pressure. This approach has been extended to the bonding of composites on steel in some studies by de Barros [88] in the case of bonded CFRP on steel pipe for mode I failure prediction and critical toughness measurement.

<figure><img src="image_56.png" /><figcaption>Figure 1.4-19: Blister test sample.</figcaption></figure>

#### 1.4.6.2. In Mode II

To compensate the issues of the ENF test, the study and development of other mode II test was made.

As exposed, the ENF test is prompt to uncontrolled crack propagation. An alternative mode II configuration, was proposed by Martin et al. [96], called the four-point end notched flexure (4ENF) test (Figure 1.4-20). This test is known as having a much more stable crack propagation so that CCM can be more commonly applied for the fracture toughness computation as described in [97]. The 4ENF test use a four-point flexure frame allowing similar rotation than the ENF test. However, this test, because of the crack being located between the two-load application points, is known as having issue related to overestimation of the critical toughness because of friction energy dissipation [91].
<figure><img src="image_57.png" /><figcaption>Figure 1.4-20: 4ENF sample.</figcaption></figure>

The End Loaded Split (ELS) test has been lately pushed forward. This set-up allows avoiding friction between the upper and lower beams along the crack length. Wang et al. [98] argued that this test is the most suitable for measuring the strain energy release rate curve, which is justified by the longer allowable crack length which help to have a stabilized crack propagation. Blackman et al. [99] asserted that it was especially adequate in the case of stiff and toughened adherents because of the requirement of the adherents to remain elastic during the whole test. The only real limitation of this case has been shown by [101]: a minimal sample length of few decimeters might be required to ensure a correct computation and stabilization of $G_{IIc}$ with the ELS samples. Figure 1.4-21 presents the overall geometry of the ELS test.

<figure><img src="image_58.png" /><figcaption>Figure 1.4-21: ELS sample.</figcaption></figure>

ELS test is based on cantilever beam geometry. The two adherents are bonded together up to the point of the initial crack length (pre-crack). The applied load generates shear in the bondline around the crack position inducing mode II loading in the adhesive. In [13] da Silva et al. add the specification that the crack propagation should occur before large displacement takes place (before the maximum deflection of the beam is equal to 0.2 times the length of the sample). This should be the case, as described in [98], when the crack length is longer than the half length of the sample.

An alternative test might be used, the single lap shear test, that corresponds to a traction test on two bonded adherents with relatively long lap length compared to other samples dimensions (Figure 1.4-22). It is the easiest test to perform to obtain energy value. Yet, as raised previously, peel stress may exist in such a geometry that cannot always be neglected [13]. In some particular case, such as CFRP (Carbon Fiber Reinforced Polymer) bonded on concrete such as in [101], the peel effect remains limited, and the single lap shear tests can be used to obtain cohesive laws [102]. For this method, one adherend must present a great stiffness compared to the other.

<figure><img src="image_59.png" /><figcaption>Figure 1.4-22: Single lap test.</figcaption></figure>
1.4.6.3. Mixed mode

The Spelt test has been developed by Fernlund et Spelt [103]. Samples geometry is closed to the DCB and ENF ones as shown in Figure 1.4-23. Different loads can be achieved by changing the main distances on the test frame, thereby varying the mode-mixity. Changing the distances generates differential load applied to the upper and lower adherends. The main advantage of this test is that it also allows the realization of pure mode tests, DCB for mode I and the ENF for mode II.

<figure><img src="image_60.png" /><figcaption>Figure 1.4-23: Spelt test.</figcaption></figure>

Another test for mixed mode has been lately developed by Sørensen et al. [101], the Double Cantilever Beam with Uneven Bending Moments (DCB-UBM). As shown in Figure 1.4-24, the principle of this test is to generate two different bending moments at the free edges of the sample. The mixed mode ratio can be changed simply by changing one transverse arm length that modifies the applied bending moment. The main advantages of the DCB-UBM test is the direct analytical computation of the J-integral which is independent of the crack length [101]. The authors showed good correlation between the obtained critical value of the J-integral and literature results in the case of a GFRP beam bonded with a thick ductile adhesive (polyurethane adhesive, 10mm-thick). Another advantage is the absence of shear forces, reducing the energy lost by friction between the opening faces

<figure><img src="image_61.png" /><figcaption>Figure 1.4-24: DCB-UBM test frame.</figcaption></figure>
Other configurations have also been used. A promising one using modified Arcan specimens [54] has been tested with various types of bonded interface (CFRP, GFRP, steel) (Figure 1.4-25).

<figure><img src="image_62.png" /><figcaption>Figure 1.4-25: Arcan test.</figcaption></figure>

The mixed mode ratio is controlled by the angle of loading in the Arcan sample from pure mode I to pure mode II. Choupani [54] was able to obtain Mode I, Mode II and mixed mode critical energy release rate values for crack propagation in the adhesive and perform numerical analyses of the experiments. He concludes that, for composite to steel bonding, peel tends to propagate the crack adhesively while shear tends to propagate the crack cohesively with a much higher critical toughness.

## 1.5. Conclusion

The objective of this bibliography was to identify the main methods to design structural adhesive joints. These methods should take into account the specificity encountered with bonded joints and propose a modeling approach with some failure criteria.

The review of the analytical models allowed highlighting the main mechanical phenomena associated with adhesively bonded joints (influence of the overlap length, existence of a characteristic anchorage length, peeling and shear concentrations at the free edges), and the main points requiring special attention in the design. This analysis also permits to identify the limits of these models for complex patch geometries and the need to use numerical approaches to obtain more consistent stress and strain fields (or energies) needed for the design of a bonded patch. Among the identified numerical tools, the finite element model seems to be the most convenient, given that the singularity effect can be considered.

The study of the available design criteria highlights the difficulties linked to their application to bonded reinforcement. Continuous failure criteria, in the elastic case, encounters some limits due to the singularities. The use of fracture mechanics approaches, on the other hand, needs to define a pre-existing initial defect length to be applicable. In the case of ductile adhesive, it seems that continuous mechanics may be used but the criteria will have to be properly defined choosing for example a critical strain approach. One approach therefore proposes to adopt coupled stress energy criteria to overcome these difficulties.
Several authors have also proposed a new approach combining both the modeling and the failure criteria of the bonded joint: the cohesive zone model. This approach couples the continuous mechanics approach as a stress criterion may be used (as the maximum stress of the cohesive law), as well as the fracture mechanics characteristics (the critical energy release corresponding to the area under the curve of the cohesive law). It is already implemented in finite element modeling and several approaches exist regarding its use in 3D.

The main methodologies used to determine the critical energy release rates in the mode I, mode II and mixed mode were then presented. It has been shown that a fine apprehension of adhesion mechanics and strength is critical for a good understanding of interface properties of bonding. Different critical toughness computation methods have been detailed such as the CCM and the MBTM. These methods propose relations between the strain energy release rate and the crack length of the samples through classical fracture mechanics investigations. However, issue to precisely monitor of the crack propagation may occur during these tests [87]. This is more deeply studied in the next chapter.
# Chapter 2: A new methodology for fracture mechanics investigations on adhesively bonded joints relying on the use of distributed optical fiber sensor

The aim of this chapter is to present a new methodology based on the use of strain distributed optical fiber sensor for the characterization of bonded assemblies. This methodology has been developed to measure relatively precisely the crack propagation during fracture mechanics tests. First, the motivation of this development is discussed. Then, the experimental investigations done for the validation of this new methodology are detailed for ENF, DCB and MMB tests.

## 2.1. Justification of the development of the new crack propagation measurement methodology

According to the equations given in the standards ([87] and [104] for DCB and MMB tests), critical toughness is directly correlated to the crack length. As consequence, a precise measurement of the crack propagation during the entire test period is necessary to ensure a correct critical toughness determination.

Standards propose a visual monitoring (with a microscope) of the crack front position on one flank of the sample during the test. However, the measurements obtained by this technique are highly dependent on the operator. In the literature, automatization of the crack propagation measurement by DIC (Digital Image Correlation) is tested to improve the accuracy measurement. Nevertheless, this method can ensure good results if the crack propagates slowly with a straight front perpendicular to the direction of propagation. In some cases, depending on the mechanical and geometrical characteristics of the bonded assembly as well as the loading, the crack front shape may not be straight in the plane perpendicular to the direction of the crack propagation. In consequence, the measurement from the flank of the specimen can lead to underestimate of several millimeters the real crack propagation. The calculation of critical toughness from such biased measurements may therefore have a large uncertainty. Other methods have been developed lately, such as using acoustic emission to measure crack length in fracture mechanics tests [105]. However, limited precision (few millimeters) remains an issue for the critical toughness computation. Table 2.1-1 summarizes the advantages and the limitations of the different techniques mentioned above for crack propagation measurement.

For the reasons mentioned above, the use of alternative measurement techniques is required to ensure a precise, robust, with high spatial resolution and easy to use, crack front measurement. In 2011, Bernasconi et al. [106] has investigated the use of an array of optical fiber sensors. The array was composed of 5 Fiber Bragg Gratings (FBGs) of 6 mm long and positioned at 21 mm intervals on the side of a joint. With this instrumentation, he showed that it is possible to detect and monitor the crack propagation in a bonded joint during a fatigue test. The methodology consisted in monitoring strain variation for each FBG. Indeed, when a crack appears and propagates near the FBG, large variation of the strain may generally occur and as consequence, the crack can be detected and localized. However, this instrumentation relying on FBG for crack front measurement presents two limitations. First, the accuracy of the position
of the crack is limited by the length of the Fiber Bragg grating. The second limitation is that several Fiber Bragg gratings must be used to obtain different crack propagation measurements.

More recently, in 2017, Meadows et al. [107] has proposed to use high spatial distributed optical fiber sensor. As for the methodology based on FBG to detect the crack position, it consisted in monitoring strain variation due to the crack propagation. Yet, distributed optical fiber allow using the whole length of the fiber as a sensor. This means that the crack propagation can be monitored continuously (if the speed of propagation is sufficiency small compared to the frequency acquisition of the system). In the study of Meadows et al., a polyimide optical fiber was imbedded in a composite sample (Carbon Fiber Reinforced Polymer) around crack tip and an ODiSI-A was used for the measurements. This system has a maximum sample rate of approximately 1 Hz and, for the post-processing of data, the gauge length and spacing were fixed to be equal to 1 mm. The comparison between the true crack lengths (measured by a microscope) and those measured with the optical fiber showed a difference less than 3 millimeters for DCB and ENF tests.

The good results of these study and their limited application case to composite sample with embedded fibers (embedded in one ply of the composite in small thickness sample, difficult and fragile samples) has motivated the idea to extend this analysis to larger application and other materials to be able to follow crack propagation in all the different case of standardized fracture mechanics tests while simplifying the measurement process. More precisely, we aimed at extending the field of application of such sensors to the field of adhesive joint, for less favorable cases (thicker samples and bonding of the fiber on the surface of the adherents) and with more recent optical fiber technology. In our case, standardized fracture mechanics tests such as DCB, ENF and MMB tests will be tested with this new methodology.

<table><caption>Table 2.1-1: Comparison (advantage and limitations) of several techniques for crack front measurement.</caption><thead><tr><th></th><th>Visual measurement</th><th>DIC measurement</th><th>Optical fiber</th><th>Acoustic emission</th></tr></thead><tbody><tr><td>Operator dependent</td><td>Yes</td><td>No</td><td>No</td><td>No</td></tr><tr><td>Automatization</td><td>Difficult</td><td>Yes</td><td>Yes</td><td>Yes</td></tr><tr><td>Position of the measured crack front</td><td>Border</td><td>Border</td><td>Center and border</td><td>Center and border</td></tr><tr><td>Crack length monitoring</td><td>Few centimeters</td><td>Few centimeters</td><td>Few meters</td><td>Few centimeter</td></tr><tr><td>Precision</td><td>&lt;1mm</td><td>&lt;1mm</td><td>1-2mm (expected)</td><td>1-5mm</td></tr></tbody></table>

## 2.2. Optical fiber crack front measurement methodologies

To determine the crack front position from strain profile measurement obtained with the optical fibers, a correct understanding of the expected strain state on the surface of the tested samples is required for mode I, mode II and mixed modes.
### 2.2.1. Theoretical ENF crack front measurement

The ENF test (mode II) corresponds to a three-point bending test (Figure 2.2-1). Upstream from the crack front (on the right in scheme), the ENF sample corresponds to a beam of thickness $2h_s + h_a$ (with $h_s$ and $h_a$, the steel and adhesive thickness respectively), while downstream from the crack front (on the left in scheme), the sample is equivalent to the superposition of two beams of thickness $h_s$ (neglecting the friction).

<figure><img src="image_63.png" /><figcaption>Figure 2.2-1: Scheme representing strain distributions for two characteristic cross-sections into the ENF sample during loading.</figcaption></figure>

According to the beam theory, the strains $\epsilon_{xx1}$ and $\epsilon_{xx2}$, respectively upstream and downstream from the crack front, that can be measured by an outer fiber (i.e. bonded on the surface of the lower plate), follow the equations (72) and (73).

$$ \epsilon_{xx1} = -\frac{M_{fz}(x)}{EI_{hz1}}\left(\frac{h_a}{2} + h_s\right) \qquad (72) $$

$$ \epsilon_{xx2} = -\frac{M_{fz}(x) h_s}{E_s I_{hz2} 4} \qquad (73) $$

The equivalent stiffness of the bonded plate is given by:

$$ EI_{hz1} = \frac{E_s b}{12} ((h_a + 2h_s)^3 - (h_a)^3) + \frac{E_a b}{12} h_a^3 \qquad (74) $$

and the moment of inertia of the steel plate is written as:

$$ I_{hz2} = \frac{b h_s^3}{12} \qquad (75) $$

With $E_a$ the adhesive Young's modulus, $b$ the width of the sample/steel plate, $M_{fz}$ the bending moment along the x axis, $E_s$ the steel Young's modulus, $h_s$ the steel plate thickness, $h_a$ the adhesive thickness, $EI_{hz1}$ the equivalent stiffness of the bonded plate and $I_{hz2}$ the moment of inertia of the steel plate.

As shown in Figure 2.2-1, $\epsilon_{xx1}$ is maximal at the load application point, while $\epsilon_{xx2}$ is maximal at the junction of the bonded and un-bonded beams (corresponding to the location of the crack
tip). Therefore, the local maximum at the junction of $\epsilon_{xx1}$ and $\epsilon_{xx2}$ is set as the position of the crack front.

This analysis relies on two important assumptions:

- the supposed plasticity/damage inside the bondline (Fracture Process Zone – FPZ – size) remains limited (this assumption is very often adopted in fracture mechanics and should be verified if possible) and has limited impact on the strain measurement obtained with the optical fibers,

- the crack front is far enough (few centimeters) from the load application to ensure no interference.

Strain monitoring on top or bottom surface and determination of local maximum value's position may therefore provide a way to monitor crack propagation for ENF test.

### 2.2.2. Theoretical DCB crack front measurement

The same analysis has been performed with the DCB test by considering an optical fiber bonded on the top and bottom surfaces (no embedded fibers). The geometry of the DCB test is reminded in Figure 2.2-2.

Following the equation (76), the maximal local rotation variation $(\frac{d\omega}{dx})$ of the samples is supposed to occur closed to the crack tip, and thus the position of the maximal strain measured at the surface of the adherent should correspond to the position of the crack front.

$$ \epsilon_{xx3} = \frac{d\omega}{dx} = -\frac{M_{fz}(x)h_s}{E_s I_{hz2}} \quad (76) $$

<figure><img src="image_64.png" /><figcaption>Figure 2.2-2: Strain into the DCB sample during loading.</figcaption></figure>

Using an outer optical fiber (positioned on bottom or top surface) should then be sufficient to determine the crack front position.

### 2.2.3. Theoretical MMB crack front measurement

The MMB test can be analyzed as the sum of an ENF and a DCB test. Upstream from the crack front (on the right in Figure 2.2-3), the MMB sample corresponds to a beam of thickness $2h_s + h_a$ while downstream from the crack front (on the left in Figure 2.2-3), the sample is equivalent to the superposition of two beams of thickness $h_s$.
According to the beam theory, the strains $\varepsilon_{xx4}$, $\varepsilon_{xx5}$ and $\varepsilon_{xx6}$ shown in Figure 2.2-3 follow the equations (77) to (79).

$$ \varepsilon_{xx4} = - \frac{M_{fz4}(x) h_s}{E_s I_{hz2}} \quad (77) $$

$$ \varepsilon_{xx5} = - \frac{M_{fz4}(x) h_a}{EI_{hz1}} \left( \frac{h_a}{2} + h_s \right) \quad (78) $$

$$ \varepsilon_{xx6} = - \frac{M_{fz6}(x) h_s}{E_s I_{hz2}} \quad (79) $$

With $M_{fz4}$ the bending moment along the x axis generated by the load $F_4$, $M_{fz5}$ the bending moment along the x axis generated by the load $F_5$ and $M_{fz6}$ the bending moment along the x axis generated by the load $F_6$, $E_s$ the steel Young's modulus, $h_s$ the steel plate thickness, $h_a$ the adhesive thickness, $EI_{hz1}$ the equivalent stiffness of the bonded plate given by the equation (74) and $I_{hz2}$ the moment of inertia of the steel plate (75).

The strains $\varepsilon_{xx4}$ and $\varepsilon_{xx6}$ are maximal at the junction of the bonded and un-bonded beams (corresponding to the location of the crack tip) while $\varepsilon_{xx5}$ is maximal at the load application point. As $M_{fz4}$ and $M_{fz5}$ have a similar effect, higher strains are generated on the top surface of the sample at the crack tip in comparison with the bottom surface ($M_{fz5}$ and $M_{fz6}$ have opposite effects). The disposition of the strain measurement instrumentation on the top surface is therefore recommended. Therefore, the local maximum at the junction of $\varepsilon_{xx4}$ and $\varepsilon_{xx5}$ is set as the position of the crack front.

<figure><img src="image_65.png" /><figcaption>Figure 2.2-3: Theoretical outer strain profiles on the lower and upper surface of an MMB sample.</figcaption></figure>

It is worth pointing out that the expected strain measurements are higher for MMB test than for the ENF test. Therefore, if the results of the ENF test are satisfying, using an outer optical fiber on the top surface, it should be sufficient to monitor the crack front position in the case of the MMB test.
## 2.3. Description of the experimental protocol

### 2.3.1. Studied samples

For mode I (DCB test), mode II (ENF test) and mixed mode (MMB test) investigations, it has been decided to use a custom geometry for the specimen, closed to the one given in the ASTM D7905 [104] standard related to the ENF test. The sample geometry has been fixed at 500x50x11mm³ and the substrates have been changed by steel plates. These dimensions, especially the thickness, have been chosen to assess the capabilities of the developed methodology on relatively thick (5 mm) and rigid adherents more representative of the considered application.

The initial crack length has also been increased compared to the ASTM standard to ensure that the crack propagation remains under control. Too little initial crack length can indeed tend to uncontrolled crack propagation for the ENF test as shown by Allix [109] or by de Barros [110]. The initial crack was made using a Teflon insert in contact with the steel adherent.

Steel surfaces were sanded and degreased prior to bonding. The bonding operation for all the specimens was processed as follows: a 1 mm steel plate was inserted at each end of the sample (out of the bonded zone); the resin was applied on the 2 surfaces prior to bonding; then the samples were assembled and pressured up to the point where the joint was 1 mm thick (the two adherents being in contact with 1 mm steel inserts). The sample was then cured at 20°C during seven days before being tested. Thickness measurements were made at several locations of the sample before testing and small variations, less than <10% of the thickness of the bondline, were seen along the length of the sample.

The adhesive used for the bonding of the sample is a cold curing epoxy (silicon toughened epoxy), called resin A. The main properties of the adhesive are given in Table 2.3-1 (from technical data sheet).

<table><caption>Table 2.3-1: Resin A mechanical properties in tension.</caption><thead><tr><th>Young modulus</th><th>Tensile strength</th><th>Strain at failure</th></tr></thead><tbody><tr><td>11 GPa</td><td>25 MPa</td><td>1%</td></tr></tbody></table>

The test frames for DCB, ENF and MMB tests were also inspired from the ones provided in the ASTM standards. The overall geometry and size were adapted to fit to the test sample geometries which are bigger than the ones in the standard. For the DCB and MMB test frame, the bonded tabs were increased in surface and thickness to avoid any debonding during the test.
### 2.3.2. DCB, ENF and MMB test bed geometries

This part presents the DCB, ENF and MMB test beds that were designed especially for the specimens described previously. The loading was carried out at a constant speed of 2 mm/min for each test.

The experiments were performed using customized devices on a Losen (300kN) testing machine. Since the expected loads were small, a secondary 5kN or 10kN load cell was used in order to provide a more accurate load measurement.

For the DCB test (Figure 2.3-1):

- the upper hinge was supported on a pull rod that had a greased loose bolt which allows to sample to self-align along the machine loading axis,

- the crack opening displacement was measured with a laser extensometer. The laser was fixed to the hinge fixture so that any rotation of the ends of the beams did not produce an error. An image of this setup can be seen in Figure 2.3-1 with the actuator located under the specimen and the secondary load cell mounted above.

For the ENF test (Figure 2.3-2):

- the upper part of the test frame was an assembly of the load cell, the push rod and fixture on the press clamp so that the load application point should only move in the vertical direction. The load application point's displacement was measured with a laser extensometer directly fixed to the push rod,

- the lower part was a two-point support consisting in two cylinders on bearing directly fixed on the floor of the press.

For the MMB test (Figure 2.3-3) the upper part consisted in the same upper assembly as for the ENF test with a different push rod shape,

- the lever part consisted in a thick I beam with a fixed hinge at one end, one cylinder on bearings in the center, and one device in the shape of horse shoe to transfer the vertical displacement from the upper part to the lever through bearings,

- the lower part was similar to the ENF device, with the left cylinder being replaced by a fixed hinge.

<figure><img src="image_66.png" /><figcaption>Figure 2.3-1: Photo of the DCB sample and DCB test bed.</figcaption></figure>
<figure><img src="image_67.png" /><figcaption>Figure 2.3-2: Photo of the ENF sample and ENF test bed.</figcaption></figure>

<figure><img src="image_68.png" /><figcaption>Figure 2.3-3: Photo of the MMB sample and MMB test bed.</figcaption></figure>

During the tests, various physical quantities have been recorded. Applied force and displacements are monitored during the whole tests using adapted load cell (precision ± 0.145%, 2.5 to 10kN) and a laser displacement sensor (precision 10µm). In addition, two local monitoring systems are used: high spatial strain distributed optical fiber sensor (Rayleigh technology) and digital image correlation.

### 2.3.1. Monitoring system by distributed optical fiber sensor

The used optical fiber distributed sensing system is a LUNA ODiSI-B. This system is based on the measurement of the Rayleigh backscattering that occurs when light propagates in the core of an optical fiber. This phenomenon is due to tiny variations of core refractive index and any changes resulting from elongation or contraction along the optical fiber modifies this signal. The technology patented by Luna relies on the use of an OFDR (Optical Frequency Domain Reflectomer) to detect these variations by comparing two Rayleigh backscattering measurements at different states. With this device, high spatial strain distributed measurement can be obtained along the entire length of an optical fiber (up to 10 m). Indeed, strain profiles can be measured with a gauge length of 1.2 mm and a gauge pitch of 0.6 mm. Moreover, the strain profile measurement can be sampled at a maximal rate of 23 Hz. It is also important to underline that this system needs no special optical fiber as sensor. Common mono-mode optical fiber with different coatings (acrylate, polyimide ...) can be used. Consequently, different types of fiber with different coatings have been studied to study the impact of this coating on the capacity of the fiber to monitor high strain gradient with high precision [111].
2.3.1.1. Optical fiber installation

The previous analytical analysis showed that bonded optical fiber on upper surface at the center
of the sample (option 1 in Figure 2.3-4) should be sufficient to follow the crack propagation.
Yet, it is supposed that, to better follow the real crack propagation, the crack measurement
should be carried out in the center of the sample (plain strain).

Thus, several options have been studied in the case of ENF tests to be sure of the final choice
of the location of the optical fiber:

- on the flank of the sample (option 2) in the direction of the crack propagation (Figure 2.3-4),

- on the flank of the specimen perpendicular to the direction of propagation (option 3), (Figure 2.3-4),

- inside the bond line (embedded optical fiber), at the center (option 4), (Figure 2.3-4).

For specimens with no embedded optical fiber in the bondline, the optical fiber was bonded
with Araldite 2014-2 adhesive on the steel surface (Figure 2.3-5a), (Figure 2.3-5b).

<figure><img src="image_69.png" /><figcaption>Figure 2.3-4: Tested positions for the optical fiber on the sample (flank, transverse, embedded and on the top).</figcaption></figure>

<figure><img src="image_70.png" /><figcaption>Figure 2.3-5: a) Example of optical fiber application on a specimen: option 3, b) DCB specimen with surface bonded optical fiber with Araldite-2014-1.</figcaption></figure>

After initial tests which are not described here, two options (2 and 3) were not kept as crack
front measurement solutions because they did not produce satisfactory results. Indeed, the
option 2 revealed to be highly dependent on the position on the flank (near the top surface or
near the bondline) possibly leading to misinterpretation of the results. Option 3 has presented
relatively good results on the measurement of the crack front position on the border of the
sample. However, it is proved to be not suitable for DCB and MMB test. Indeed, as the crack
propagates, and the local opening increases, the measured local strains exceed the measurement
capacity of the fiber (10 000 µm/m), and the optical fiber breaks after only few centimeters of
crack propagation. The next analysis will be performed only with options 1(also called exterior
optical fiber) and 4 (also called embedded optical fiber).
2.3.1.2. Choice of the optical fiber technology

According to the theoretical analysis, the crack propagation monitoring should be possible with optical fiber through the recording of the position of the local maximal strain on the surface of the sample. The strains on the surface are transmitted to the core of the optical fiber through the used adhesive and through the coating of the fiber (Figure 2.3-6). The thickness and the material of the coating is critical on the capacity to determine high strain gradient. Thus, a thicker and less rigid coating may reduce this capacity.

An ENF test has been conducted using the previously described sample (with an initial crack length of 6 cm) to study the impact of the choice of the fiber on the strain measurement. Two different optical fibers were bonded with the Araldite A-2014-1 adhesive, centered on the surface of the sample with a 1 mm gap between them. The two studied coatings were polyimide and acrylate. The main geometrical properties of these two different fibers are summarized in Figure 2.3-6.

<figure><img src="image_71.png" /><figcaption>Figure 2.3-6: Scheme of the two studied optical fiber technologies and connection to the samples.</figcaption></figure>

Figure 2.3-7 presents the resulting strain profiles obtained during this test for one load level with the two different studied optical fibers (with polyimide and acrylate coating).

<figure><img src="image_72.png" /><figcaption>Figure 2.3-7: Local strain measurement during an ENF test with polyimide and acrylate optical fiber coating.</figcaption></figure>
The curves show that depending on the type of fibers, the results are very different. The polyimide fiber shows distinct strain peak at the load application point and at the crack front position, while the acrylate fiber results show no visible peak. This may be explained by the thickness of the coating and the rigidity of the coating or/and the coating-adhesive interface (Figure 2.3-6). The impact of the coating on the smoothing of the strain gradient is easily visible from these results. To be able to follow the crack propagation during DCB, ENF and MMB tests, it seems, in our case, that the use of polyimide coated optical fiber is absolutely essential. Next parts will present the detailed analysis of the crack position monitoring during complete DCB, ENF and MMB tests using the polyimide optical fiber.

## 2.4. DCB test investigations

### 2.4.1. Load/displacement curve analysis and failure mode

The obtained load/displacement curve for the DCB test is shown in Figure 2.4-1. Three stages can be observed. The first one is a linear elastic stage up to point A ($t_A=85s$). Once the load exceeds the point of non-linearity, damage appears in the bondline, extends and forms the fracture process zone (FPZ). This leads to a slope decrease, and an increase of the compliance $C$ ($C=\delta/P$, $\delta$ the opening at the load application point, $P$ the Load). When the FPZ is fully developed, the load/displacement curve reaches its maximum point (point B, $t_B=115s$). The crack propagation is assumed to occur after the first load decrease, since the sample cannot store more elastic energy. Consequently, the upcoming analysis will be focused on the results computed after point B. The failure mode for this test was cohesive in the bondline.

<figure><img src="image_73.png" /><figcaption>Figure 2.4-1: DCB resin A load/time curve.</figcaption></figure>

### 2.4.2. Exterior optical fiber

For the DCB test, according to the theoretical analysis, an optical fiber on the upper surface of the sample seems sufficient to measure the crack propagation. Figure 2.4-2 presents the obtained experimental strain profiles along the lap length with the optical fiber, before and during the crack propagation. Each curve corresponds to a different time. Note also that all
successive strain profiles were shifted in abscissa by the same amount to match the initial crack propagation of 85 mm.

<figure><img src="image_74.png" /><figcaption>Figure 2.4-2: DCB exterior optical fiber continuous strain measurements along lap length for different times.</figcaption></figure>

As proposed from the theoretical analysis described in the section 2.2.2, the position of the peak (red dots in Figure 2.4-2) is used to monitor the crack propagation. A limited shift of the maximum of the curve to the right can be observed at $t_A$, which is supposed to be linked to the beginning of the creation of the process zone in the bondline. The main propagation is supposed to happen after $t_B$ as visible in Figure 2.4-2. After this time, it can be shown that the maximal strain value remains stable at a value of around 1250µm/m. Thus, the local stress state remains similar at the crack tip during the crack propagation.

Figure 2.4-3 presents the crack propagation curve obtained from the optical fiber measurements. A smoothing of the data with the use of local mean has been made to reduce the raw crack front computation noise (each raw data value is converted to average of the 5 data points centered on the original value). It shows that the main crack propagation (macro-crack) occurred after the time $t_B$. The initial shift between $t_A$ and $t_B$ measures 3 mm. Two artefacts measurement can be seen at around 150 s which generate local error in the crack length measurement, but without compromising the validity of the measurement.

<figure><img src="image_75.png" /><figcaption>Figure 2.4-3: Crack propagation curve for the DCB test - data processing via local mean approximation.</figcaption></figure>
The resulting curve shows that, as visible in Figure 2.4-1, the crack has propagated in a stable manner during the whole test. The peak of maximal strain has been correctly defined during nearly the whole crack propagation. The overall crack propagation length at the end of the test was 122 mm.

### 2.4.3. Crack length visual validation

After the end of the test, a dye penetrant was used to highlight the area between the initial crack front and the final crack front (colored by the product) marking the cracked area of the sample. The measurement of the length of this area at the middle of the sample is set as the crack length at the end of the test as visible in the Figure 2.4-4a.

<figure><img src="image_76.png" /><figcaption>Figure 2.4-4: Visual measurement of the crack length of the DCB test, a) in the middle and b) at the border.</figcaption></figure>

The result shows that the final crack length obtained by the optical fiber (122 mm) is really close to the value measured with a caliper (123.1 mm). This result confirms that the exterior optical fiber is able to monitor the crack propagation for DCB test.

It is worth noting that the maximal difference between the crack front position at the center of the sample and on the edges (where visual inspection and DIC are normally carried out) is equal to 4.7 mm. This difference is important in regards with the studied propagation distance of 122 mm. This result shows clearly that the crack propagation on the flank on the specimen can lead to underestimate the crack propagation. Thanks to the distributed optical fiber, a better measurement could be achieved.

## 2.5. ENF test investigations

### 2.5.1. Load vs time curve analysis and failure mode

Two ENF tests were conducted:

- the first one with only the exterior optical fiber, to validate if the strain peak measured on the surface could be used to measure the crack front position,

- the second one, with an embedded optical fiber, to study the local strain in the adhesive and compare it to a finite element model, prior to the crack propagation.

The obtained load vs time curve for the first ENF test (test 1) is shown in Figure 2.5-1a. To limit the crack propagation speed and help to monitor the displacement of the strain peak, the loading of the sample was stopped (fixed displacement) when the maximum load was reached.
However, the crack propagation still occured as can be observed with the rapid decrease of the load. The resulting load/displacement curve can be decomposed in two stages. The first one is a linear elastic stage up to time t<sub>B</sub> = 189 s. Once the load exceeds this limit, the crack propagation has occured up to the local minimum after the first load decrease. Note that, no nonlinear loading correlated to the creation of the fracture process zone in the bondline is clearly observed for the test 1.

For the test 2, three stages can be observed: a linear loading up to time t<sub>A</sub> = 40 s, a nonlinear
loading which is correlated to the creation of the fracture process zone in the bondline, and the
crack propagation after time t<sub>B</sub> = 58 s (Figure 2.5-1b). Consequently, the upcoming analysis
will be focused on the results computed after time t<sub>B</sub>. The failure mode for this test was cohesive
in the bondline.

<figure><img src="image_77.png" /><figcaption>Figure 2.5-1: ENF resin A load / time curve, a) test 1 and b) test 2.</figcaption></figure>

**2.5.2. Exterior optical fiber analysis and crack propagation determination**

Figure 2.5-2 presents the obtained strain measurements along the length of the sample for the exterior optical fiber during the crack propagation of the test 1. Each curve corresponds to a different time. The two initial peaks described in part 2.2.1 (theoretical analysis) are clearly visible before the crack propagation. For the first peak (on the left in Figure 2.5-2), it can be observed that its position remains stable in the beginning of the loading, changed during the test (related to the crack propagation). This position corresponds to the initial crack front
position. Note that all strain profiles were shifted in abscissa by the same amount to correspond to the initial crack length of 85 mm (vertical blue line in Figure 2.5-2). When the crack propagates, the first peak shifts in direction of the load application peak point. Once the crack propagation is finished, the two peaks have merged near the position of the load application point. The strain loss on the right side of the curve is associated with the load decrease after the crack propagation.

<figure><img src="image_78.png" /><figcaption>Figure 2.5-2: Exterior optical fiber distributed strain measurements between times 154.76 s and 203.0 s.</figcaption></figure>

Figure 2.5-3 represents the maximal values of the optical fiber strain measurements versus the axial length for different times after the crack propagation. We can observe a shift of the maximum of the curve to the right, indicating a displacement of the crack tip. We can also observe that the main crack propagation occurs between $t = 189.01$ s and $t = 191.0$ s with a fast propagation between $t = 190.57$ s and $t = 191.0$ s.

<figure><img src="image_79.png" /><figcaption>Figure 2.5-3: Shift of the first peak of the strain profiles measured during the ENF test.</figcaption></figure>
<figure><img src="image_80.png" /><figcaption>Figure 2.5-4: Crack propagation curve for the ENF test – raw data processing via local mean approximation.</figcaption></figure>

Figure 2.5-4 shows the crack propagation curve obtained with the distributed optical fiber sensor. From this curve, it can be confirmed that the crack propagated very quickly at starting at about 190.6 s. Unfortunately, the frequency of the interrogator (~20Hz) was insufficient to obtain a relatively high number of points during the fast part of the crack propagation (~50cm/s). As consequence, no local meaning computation could be done. In spite of this, the final crack length can be measured by the optical fiber. A value of 84 ± 1 mm is found.

### 2.5.3. Crack length visual validation

After the end of the test, dye penetrant was used to highlight the initial and final crack front. The measurement of the length of this area at the middle of the sample is set as the crack length at the end of the test as visible in Figure 2.5-5.

<figure><img src="image_81.png" /><figcaption>Figure 2.5-5: Visual measurement of the crack length at the end of the ENF test.</figcaption></figure>

In the case of the ENF test, the result shows that the final crack length obtained by the optical fiber is 84 mm which is really close to the value measured with a caliper (86 mm). This result seems to confirm that the exterior optical fiber is able to monitor the crack propagation for the ENF test. It must be noted that, in this case, the crack front has remained straight at the end of the test. It is consistent with the results presented in [107] in the case of ENF tests realized on composite samples. In this study the crack front has been monitored during the test showing that it may change in orientation (non-perpendicular to the propagation direction), but remains always straight.