Keywords: Elastplastcity, Puck's theory, Continuum Damage Mechanics, FRPs. v Chapter 1 Composite materials and damage modeling

In the current research work, numerical modeling based on the coupled plasticity and Continuum Damage Mechanics (CDM) theory has been carried out for Fiber Reinforced Polymers (FRPs) composites. The objectives were to provide comparatively a more reliable and accurate numerical prediction tool for quasi-static loading. For this purpose, a relatively more efficient failure theory, Puck's theory, was chosen for triggering the intra-laminar meso-scale damage onset in an orthotropic homogenized ply. Being a physics based failure theory relying on Mohr-Coulomb hypothesis, it is capable to predict the meso-crack in addition to its orientation in continuous FRPs based composites. The thermodynamically consistent damage evolution law was defined to accumulate the damage leading to the subsequent stiffness degradation coupled with the isotropic hardening plasticity. The damage modeling was based on the meso-scale which is the ply-level for the Unidirectional (UD) FRPs. Even though, Puck's theory has been used before but in the present work, the prediction of the mechanical behavior and the subsequent ultimate failure has been significantly improved from the previous investigations by employing the coupled damage evolution and plasticity. The nonlinear mechanical behavior and ultimate failure were predicted for the small strain timeindependent boundary value problem. Afterwards, 3D damage model was developed based on 3D Puck's theory and was used in the indentation and adhesion wear sliding against a metallic ball. Good correlation was established between the failure mechanisms that occur in the adhesion wear in FRPs with metallic counterparts. In the second part of the thesis, UD mechanical characterization was performed and Arcan fixture was modified for testing the FRPs in combined in-plane shear and tensile loading. The effects of damage induced in pure shear state were highlighted experimentally on the tensile behavior using the optimized butterfly Arcan specimen in which a fairly uniform shear stress field was observed.

Dedication

With my sincere devotion and reverence, I dedicate this thesis to my dearest father (Dajee) for his bestowing unending good wishes and, he guided me in the pursuit of knowledge through hard work in all his life, who is no longer among us! First of all, I would like to offer my profound gratitude to all of the Jury members who have contributed to ameliorate my thesis further through their constructive remarks and valuable technical suggestions.

I shall be failing in my duty if I would not have a special mention of the unstinted support, guidance, and encouragement received from my thesis director Prof.

Stéphane Panier throughout the tenure of my PhD. He served as a beacon of light and source of inspiration for me that proved as a driving force to pass this milestone. Furthermore, I owe special thanks to him to introduce me into the continuum damage mechanics of composite materials which, at the start looked very perplexing and challenging for me, but later became highly fascinating and enticing due to his guidance and after having prolong discussion with him.

ISRAR UD DIN

Résumé étendu en français

Contexte de la thèse:

Les matériaux composites jouent un rôle de plus en plus important dans le monde technologique actuel et remplacent ainsi progressivement les métaux dans les applications structurelles. L'objectif principal de l'introduction de matériaux composites dans les structures est d'améliorer le rapport résistance/masse en plus d'autres avantages tels que la flexibilité de la conception, la réduction du nombre de pièces, la résistance à la corrosion, la résistance à la fatigue et aux chocs. En outre, le matériau composite peut être personnalisé en fonction des directions de chargement prévues. La fabrication de stratifiés à base de matériaux composites unidirectionnels (UD) pré-imprégnés présente des difficultés et présente un coût plus élevé. Pour remédier à ce problème, des tissus secs textiles présentant un coût inférieur et une cadence de production plus élevée ont été développés. De plus, ces tissus secs ne nécessitent pas de conditions de stockage contrôlées. Les tissus non ondulés (NCF) sont un type de composites textiles dans lesquels les plis sont cousus ensemble par le fil de polyester dans le sens de l'épaisseur mais sont non tissés. Pour réduire les temps de développement, des modèles numériques efficaces doivent être utilisés lors de la phase de développement du produit pour accélérer le développement tout en réduisant le nombre d'essais de qualification. À cette fin, de nouveaux modèles doivent être développés, qui devront s'appuyer sur un nombre moins important d'essais d'identification et plus faciles à réaliser.

Motivation de la thèse

La modélisation des endommagements dans les matériaux composites à matrice polymère renforcés de fibres (FRP) est une tâche complexe qui tient compte de la nature hétérogène des matériaux composites ainsi que de la nature multi-échelle de l'initiation et de la progression des endommagements. Il existe trois échelles pour le développement des modèles d'endommagement: micro, méso et macro. La modélisation des endommagements à l'échelle méso dépend de l'architecture du type de matériau composite. Par exemple, dans le cas d'un stratifié à base de couches UD, une seule couche est considérée comme l'unité de modélisation des endommagements à l'échelle méso. Dans le travail actuel, la modélisation est basée sur le niveau moyen. Par vi conséquent, la réponse mécanique au niveau macro est prédite ici par le couplage de l'échelle méso avec l'échelle macro. L'objectif principal de la thèse est de développer un outil de prévision des endommagements sur la base de la théorie continue de l'endommagement (CDM) et de la plasticité dans un cadre couplé. Le secteur industriel a absolument besoin que les modèles de prévision soient efficaces en termes de rapport complexité de calcul/temps de résolution, de précision et de caractérisation/identification du modèle. De plus, ces modèles doivent être simples et faciles à utiliser par les ingénieurs concepteurs dans un environnement industriel.

Objectifs de la thèse

L'objectif général de la thèse est de développer un modèle de d'endommagement pour prédire les performances des composites à matrice polymère et fibres continues. Les objectifs du travail de recherche sont énumérés comme suit:

1. Effectuer l'analyse des endommagements quasi-statiques en prenant en compte uniquement les modes d'endommagements intra-laminaires et développer comparativement un modèle d'endommagement plus précis que les modèles existants.

2. Étudier l'état de contrainte 3D et les modes d'endommagements rencontrés dans les FRP dans le cas d'un contact frottant et glissant d'une bille métallique sur un matériau composite s'appuyant sur un modèle d'endommagement 3D.

3. Étudier expérimentalement et numériquement le comportement des FRP en cisaillement pur et en traction / cisaillement dans le plan.

Pour atteindre ces objectifs, le manuscrit se décompose en quatre chapitres et finissant par une conclusion et des perspectives :

• Chapitre 1: Le premier chapitre présente une brève introduction générale concernant les types de composites et leurs applications dans l'industrie. Les théories sur les ruptures et les endommagements, les mécanismes d'endommagement et la nature multi-échelle des FRP sont discutés avec un accent particulier mis sur la théorie de Puck basée sur des facteurs physiques.

Enfin, un cadre est établi pour la modélisation numérique basée sur la théorie de l'endommagement couplé avec la plasticité.

• Chapitre 2: Ce chapitre comprend une étude bibliographique détaillée sur les modèles d'endommagement basés sur la mécanique continue de l'endommagement. Un modèle 2D en contrainte plane est proposé et implémenté dans ABAQUS via la subroutine UMAT. Les résultats obtenus ont été comparés à certains modèles de la littérature scientifique et à des résultats expérimentaux.

• Chapitre 3: Un modèle 3D d'endommagement avec une loi de comportement élastique a été formulé et mis en oeuvre sur la base de la théorie de 3D Puck et de la dissipation de l'énergie de rupture. L'algorithme de recherche du plan critique de rupture de Puck a été implémenté dans l'UMAT. Les effets unilatéraux et la récupération de la rigidité sont discutés en cas d'inversion du chargement en utilisant le concept de fermeture/ouverture de fissure en compression et en traction. L'indentation et le phénomène de glissement avec frottement entre une bille métallique et un matériau composite est ensuite étudié.

Les endommagements causés par le frottement sont prédits avec la théorie de 3D

Puck et comparés avec les observations expérimentales.

• Chapitre 4: Dans la première partie du chapitre 4, la caractérisation des matériaux est effectuée conformément aux normes ASTM applicables. Par la suite, des essais dits Arcan ont été menés dans le cas du cisaillement pur et du chargement bi-axial traction/cisaillement dans le plan. Pour la réalisation de cette partie, un nouveau dispositif expérimental Arcan a été conçu et fabriqué.

Une optimisation topologique de l'éprouvette papillon de l'essai Arcan a également été réalisée par le biais de scripts Python dans l'environnement ABAQUS. L'optimisation visait à obtenir un champ de contrainte de cisaillement relativement uniforme dans la partie utile de l'éprouvette. Outre le test de cisaillement pur et les tests bi-axiaux, le nouveau dispositif expérimental a également été utilisé pour mettre en évidence les effets d'endommagements diffus induits en cisaillement pur sur le comportement en traction des FRP. De plus, les résultats expérimentaux en cisaillement pur ont été complétés par des simulations numériques en utilisant le modèle développé au chapitre 2.

• Chapitre 5: Le dernier chapitre traite des résultats et des conclusions concernant les domaines numérique et expérimental de la présente thèse. En outre, les viii limites du modèle sont discutées et des suggestions ont été formulées pour les travaux futurs.

CHAPITRE 1

Le but de ce chapitre est de fournir une introduction aux matériaux composites à matrice polymère renforcés par des fibres continues et à leur classification ainsi que l'importance des matériaux composites dans le monde industriel actuel. Ensuite, les tissus non tissés (NCF) sont discutés en plus de leurs applications industrielles et leur fabrication. Les modes d'endommagement et leur nature multi-échelle sont ensuite présentés avec un accent particulier mis sur la modélisation des endommagements à l'échelle mésoscopique. Un examen des théories de rupture, en particulier de la théorie de Puck, et des méso-modèles est également présenté.

Le programme mondial sur l'étude des ruptures des composites (WWFE) ainsi que d'autres études présentées dans ce chapitre ont montré que le critère de Puck était l'un des meilleurs critères pour prédire la rupture des composites stratifiés à matrice polymère.

Des articles récents révèlent que ce critère peut également être utilisé comme un bon indicateur pour détecter l'initiation de l'endommagement à l'échelle mésoscopique. De plus, l'orientation de la fissure et son degré de nocivité peuvent également être distingués en raison du fondement physique sur lequel s'appuie la théorie de Puck. Un autre point intéressant de cette théorie est l'utilisation du même nombre de paramètres de résistance que ceux requis par d'autres théories conventionnelles. Cinq paramètres en résistance au total sont nécessaires pour la théorie de Puck. Par conséquent, la théorie de Puck est le meilleur candidat à utiliser dans le cadre de la mécanique continue de l'endommagement. Par conséquent, il est proposé de développer un modèle d'endommagement dans le chapitre suivant, basé sur la théorie de Puck dans le cas des contraintes planes.

CHAPITRE 2

Ce chapitre donne une description détaillée de la modélisation des endommagements anisotropes sur la base de la théorie de l'endommagement des milieux continus couplée ix à la plasticité. La loi d'évolution des endommagements cohérente d'un point de vue thermodynamique est définie pour accumuler les endommagements conduisant à la dégradation progressive de la rigidité associée à la plasticité à écrouissage isotrope. Le modèle utilise une fonction d'écrouissage isotrope à un paramètre et ne possède aucune sensibilité à la contrainte hydrostatique. La méthode des éléments finis (FEM) est utilisée pour prédire le comportement mécanique dans des conditions de chargements quasi-statiques. L'initiation de l'endommagement et l'évolution sont formulées en tenant compte de la version 2D de la théorie de Puck. Le modèle est basé sur les valeurs de résistance requises pour un pli et les énergies de rupture en plus des paramètres du potentiel plastique.

Une loi de comportement 2D sur les endommagements élastoplastiques anisotropes est donc développée. L'initiation de méso-dommages intra-couche, la rupture de la fibre (FF) et les ruptures inter-fibre (IFF) peuvent être prédits.

Un maillage approprié a été utilisé dans les simulations MEF en gardant à l'esprit son influence sur les résultats prévus basés sur l'étude de sensibilité du maillage. La capacité de prédiction non linéaire de la loi de comportement indépendante de la vitesse a été validée en comparant les résultats expérimentaux obtenus avec les stratifiés à fibres de carbone composés de deux angles: [±30°] 2s et [±45°] 2s . Ces stratifiés ont été choisis à dessein en raison de leur comportement non linéaire, en particulier ceux avec l'empilement à [±45°] 2s .

Les résultats obtenus sont excellents par rapport à ceux obtenus avec d'autres approches.

Ensuite, la charge à la rupture ultime pour des éprouvettes trouées constituées de fibre de carbone associées à une résine époxy a été prédite pour certains des stratifiés choisis dans la littérature. Afin de ne pas prendre en compte le délaminage, une adhésion parfaite était supposée entre les plis et seuls les essais sans délaminage ont été considérés.

CHAPITRE 3

Dans ce chapitre, la théorie de 3D Puck a été de nouveau utilisée dans le cadre de l'endommagement des milieux continus. L'approche de Puck s'appuyant sur la détermination d'un plan critique de rupture (dans lequel le critère est exprimé), un travail important d'implémentation d'algorithme de recherche de ce plan dit critique a été mené.

La détermination de l'angle de rupture (plan critique) a été démontrée et validée sur des cas académiques.

x De plus, les effets unilatéraux et la récupération de la rigidité dans les inversions de chargements ont été modélisés. L'augmentation de la résistance au cisaillement lorsqu'on augmente la contrainte de compression transversale a été réalisée en traçant l'enveloppe de rupture et comparée à celle de la théorie de Hashin.

Il existe de nombreuses applications dans lesquelles l'influence des composantes de Des tests de cisaillement purs ont été effectués en contrôle de déplacement pour les deux types d'éprouvette. La réponse contrainte-déformation de cisaillement a été observée presque similaire pour les deux types de stratifiés. De plus, le comportement en cisaillement pur de l'éprouvette UD a également été étudié numériquement. En outre, les éprouvettes ont également été testées avec un chargement incliné à 45° et 90°, ce qui correspond respectivement au cas de contrainte bi-axiale avec un rapport de charge de 1 et au test de traction pure.

Enfin, les effets des endommagements induits lors d'essais de cisaillement à l'état pur sur des éprouvettes à couches croisées ont été étudiés sur le comportement en traction. Lors de ces essais, l'échantillon a d'abord été chargé en cisaillement jusqu'à une charge prédéfinie inférieure à la charge de rupture finale. Ensuite, la même éprouvette a été démontée et tournée à 90° dans le dispositif Arcan. Il a été observé que les endommagements générés par cisaillement ont dégradé le comportement en traction du stratifié.

General Introduction

Context

Composite materials play an important role in the current state-of-the-art technological world, and thus replacing the metals progressively in the structural applications. The main purpose of the introduction of composite materials into the structures is to enhance the strength-to-weight ratio in addition to other benefits such as design flexibility, less number of parts, corrosion resistance, higher fatigue, and impact resistance. Besides, the material can be tailored according to the anticipated loading directions. The manufacturing of prepregs UD based laminates possess difficulty and add higher cost to the products. To address this issue, textile dry fabrics are introduced that has less cost and higher production rate as compared to the UD based laminates. In addition, these dry fabrics do not need controlled storage conditions. Non Crimp Fabrics (NCFs) is a type of textile composites in which the plies are stitched together by the polyester yarn in thickness direction. This increases the deposition rate considerably. To get further economy, efficient numerical models shall be used in the product development phase to make development faster and at the same time reduce the number of qualification tests.

For this purpose, new models shall be developed that should rely on less number of easier identification tests.

Motivations

Modeling of damage in Fiber Reinforced Polymers (FRPs) is a complex task keeping in view the heterogeneous nature of composite materials in addition to the multi-scale nature of the damage development and progression. There are three scales for the development of the damage models: Micro-, meso and macro-. Meso-scale damage modeling depends on the architecture of the type of composite material. For instance in case of UD plies based laminate, a single lamina is considered as the unit for meso-scale damage modeling. In the current work, modeling is based on the meso-level. Hence, the macro-level mechanical response is predicted here by the coupling of meso-scale with the macro-scale. The main motivation of the thesis is to develop a damage prediction tool on the basis of the theory of Continuum Damage Mechanics (CDM) and plasticity xiii in a coupled framework. It is the dire need of the industry that the prediction models shall be efficient in terms of computational complexity/solution time, accuracy and as well as characterization/identification of the model. Furthermore, these models shall be simpler and easy to be used by the design engineers in the industrial environment.

Objectives

In general, the aim and scope of the thesis are to develop a damage model for predicting the performances of the FRPs. Objectives of the research work are summarized as follows:

1. To perform the quasi-static damage analysis by considering only the intralaminar damage modes, and develop comparatively a more accurate damage model as compared to the previous damage models.

2. To study the 3D state of stress and the damage modes encountered in FRPs under indentation and sliding against the metallic ball on the basis of a 3D damage model.

3.

To investigate experimentally and numerically the behavior of FRPs in pure shear and combined in-plane shear/normal loading.

Outline

Chapter 1: The first chapter presents a brief general introduction regarding the composite types and their applications in the industries. Failure theories, failure mechanisms and the multi-scale nature of the FRPs are discussed with special emphasis is given to the physically based Puck's theory. Lastly, a foundation is made for the numerical modeling based on the coupled damage and plasticity theory in the framework of Continuum Damage Mechanics (CDM).

Chapter 2: This chapter comprises the detailed literature review on previous damage models and prediction based on the theory of CDM for irreversible processes. Based on the comprehensive literature review, a plane stress model was proposed and implemented in ABAQUS via UMAT. The results predicted were compared with some previous models and the experiments there-in which showed good accuracy.

Chapter 3:

Here a 3D damage model with elastic constitutive law was formulated and implemented based on the 3D Puck's theory, and the fracture energy dissipation. The xiv algorithm for the Puck's fracture plane searching was included in the UMAT.

Furthermore, the unilateral effects and stiffness recovery is discussed in case of load reversals that utilizing the crack closing/opening concept in compression and tension loads. Eventually, the indentation and sliding phenomenon against the metallic ball and the damage caused by the sliding is modeled with the 3D Puck's theory.

Table of Contents

Résumé [START_REF] Simo | Consistent tangent oprators for rate-independent elastoplasticity[END_REF]..........

2.3

List of Figures

List of Tables

Introduction 1.1

The goal of this chapter is to provide an introduction to Fiber Reinforced Polymers (FRPs) materials and its classification. It commences with highlighting the importance of the composite materials. Then, the Non Crimp Fabrics (NCFs) are discussed in addition to its manufacturing and industrial applications. Damage modes and multiscale nature of the damage phenomena in FRPs is elaborated with special emphasis is given to the meso-scale damage modeling. A review of the failure theories, specifically the Puck's theory, and meso-models for UD FRPs are also presented.

Importance of composites 1.2

The excellent mechanical properties have heralded composite materials a basic necessity in the latest generation of the passenger aircraft, such as the Boeing B-787 and Airbus A-350, where the composite materials account for more than 50% as portrayed in Figure 1-1.

Figure 1-1: Materials distribution in Boeing B-787 [START_REF] Campbell | Manufacturing Technology for Aerospace Structural Materials[END_REF] The application of composites in the aerospace structures and its increase with time is depicted in Figure 1-2 which describes that how composites are continuously replacing other materials in aerospace sector [START_REF] Kassapoglou | Design and analysis of composite structures: with application to aerospace structures[END_REF]. A similar trend can also be found in other industrial sectors such as automobiles, energy, and sports goods, etc.

(Figure 1-3).

Composites and its classification 1.3

Generally, composites can be classified based on the type of reinforcement and their topological configuration. The reinforcements may be in particulate form such as gravel in the concrete, or in continuous form as carbon fiber or glass fiber, or the reinforcement may be discontinuous. Collectively, composites which are reinforced by either continuous or discrete fibers are known as Fiber Reinforced Polymers (FRPs) which are further classified into Uni-Directional (UD) prepregs laminated composites and textile composites. UD laminated composites are the conventional pre-impregnated composites in which the partially cured plies are stacked together in order to fabricate the final desired shape. One of the drawbacks of the UD prepregs based FRPs composites is the higher and laborious fabrication cost encountered by almost all types of the industrial sectors. For this purpose, alternatives such as textile preform fabrics have been introduced keeping in mind the same structural benefits coupled with cost efficient, easy, and automatic production methods. Textile composites are fabricated from the dry preforms such as woven fabrics and stitched fabrics for example as Non Crimp Fabrics (NCFs). With the usage of these dry textile preforms, the storage in a controlled environment has also been obviated which is a necessary requirement in case of UD prepregs. Structural composite is also one of the types of composite materials. Different materials are stacked on each other in structural composites: for instance, the sandwich panel which comprises of two skin layers and a core is one of the examples of structural composite. The skins may be either fiber reinforced or some other suitable material like metal, ceramic, wood depending on the intended application.

Textile composites

Previously, it was mentioned that the manufacturing and material cost of FRPs is one of the disadvantages in the current industrialized high volume production of UD prepreg based laminates. Hence textile composite, which avail different fiber topologies, is a response to address this issue, due to the reduced manufacturing and material cost. Such materials are optimum to be used in the structures where a combination of intermediate level of mechanical properties with low-cost is the objective of the design [START_REF] Ernest G | Multiscale progressive failure analysis of textile composites[END_REF]. Textile composites consist of different types which are classified with respect to the internal architecture of the fiber yarns and the way these yarns are interlaced with each other like braided, woven, and stitched (NCFs) as illustrated in Figure 1-4 [START_REF] Lomov Sv | Textile Composite Materials: Polymer Matrix Composites[END_REF]. Although NCFs have many types in regard of the consolidation or binding of the plies in through-thickness direction, but the present thesis will only discuss the NCFs composites in which the preforms are stitched together with the help of polyester yarns.

The function of these polyester yarns is only to keep the dry preform intact. Other ways of through-thickness binding are the z-pining (Mouritz AP, 2010), and bonded preforms. NCFs and their types are discussed in great detail in [START_REF] Lomov Sv | Textile Composite Materials: Polymer Matrix Composites[END_REF]. As far as the fiber orientation is concerned, NCFs based laminates consist of bi-axial or biangle, tri-axial, and or quadri-axial. The through-thickness yarn and the lay-up of a typical quadri-axial NCF laminate are shown in Figure 12345. The through-thickness stitching affects the permeability (flow of resin during fabrication) and the mechanical properties, but it reduces the cost and increases the productivity significantly. NCFs based composites offer lower fabrication cost, higher deposition rate with approximately same or little lower in-plane mechanical properties in comparison to that of UD based prepreg laminates. In addition, it has enhanced the sustainable out-of-plane properties such as impact resistance and delamination than the UD prepreg laminated composites but inferior than the woven textile composites [START_REF] Leif E | Local models for NCF composites materials mechanical performance prediction[END_REF]. To cater the need of the industries, new types of fabrics are always introduced into the market for instance bi-angle NCF which constitutes of laminates having plies along the two orientations only across the whole laminate.

Bi-angle NCFs 1.4

Ideally speaking, NCFs laminates consist of the straight fibers without any crimping or undulation of the fiber yarns which result into enhanced in-plane mechanical properties as compared to the inferior in-plane properties of rest of the textile composites.

Comparing with the UD prepreg laminated composites, the through-thickness stitching improves the ease-in-handling, higher drapeability (transformation into complex shapes) and also the inter-laminar mechanical properties like fracture toughness, and impact resistance [START_REF] Beier U | Mechanical performance of carbon fiber-reinforced composites based on stitched preforms[END_REF]. Therefore, NCFs based composites have a very bright future prospects where intermediate level of in-plane mechanical performance is desirable with low cost [START_REF] Edgren | Formation of damage and its effects on non-crimp fabric reinforced composites loaded in tension[END_REF].

Classical Quasi Isotropic (QI) laminates based on the black aluminum design concept are useful where (i) the loads are not very well understood or (ii) the loads are very complex and multi-directional in nature for example in the ocean energy devices as tidal turbines [START_REF] Kennedy Cr | A multiaxial fatigue damage model for fibre reinforced polymer composites[END_REF]. In contrary, when the loads are directional or one-axis then the new laminate design concept of bi-angle is advantageous to follow. The biangle NCFs laminate is a new concept developed by the Tsai group and Chomarat in which, instead of the conventional laminate design of four angles (Quasi-isotropic, QI), two angles are incorporated across the whole laminate [START_REF] Cognet M | A totally new concept of a laminated composites--The amazing bi-angle thinply NCF[END_REF]. Laminates which make use of only two angles are thus become unbalanced and highly anisotropic for instance: [0/25°] or [0/45°]. Using minimum of 16 bi-angle fabrics or 32 plies of the novel thin NCFs patented by Tsai group having 150 GSM per fabrics, produce a homogenized laminate which does not warp regardless the laminate is symmetric or unsymmetric. Hence all these new advancements give an enormous relief and freedom to the laminate designers and manufacturers.

Applications of shallow bi-angle NCFs

Bi-angle NCFs tapes are available in the market in stitched forms of two plies which are fabricated by a new technology known as tow-spread thin-ply technology originally developed at Japan by [START_REF] Kawabe K | New technology for opening various reinforcing fiber tows[END_REF]. Being unbalanced and unsymmetric [0/25°], [0/45°], NCFs laminates are best suited in the slender structures where there is combined loading of bending-twisting in action [START_REF] Tsai | Weight and cost reduction by using unbalanced and un-symmetric lamiantes[END_REF]. Some of the examples include, wings, stabilizers, turbine fans, wind turbine or helicopter rotor blades, propellers, shafts, golf sticks, and tennis rackets (Arteiro A, 2013). Figure 123456contains some of the slender structures where NCFs and thin plies based laminates can be efficiently installed.

The mechanical performance in tension and bending of shallow bi-angle thin-ply laminates has been investigated in [START_REF] Kim Yhn | Effects of shallow bi-angle thin-ply laminates on the structural performance of composite wing[END_REF] : a high resistance to delamination and lower matrix micro-cracking has been attributed to the lower thickness of the plies in the experimental demonstration. The shallow bi-angle laminates of [±25°] have resulted 329% higher x-directional elastic modulus at the cost of only 25% reduction in the transverse direction modulus and 8% drop in the shear modulus. As a consequence, it is strongly recommended by the authors to use shallow angles in the slender structures.

Manufacturing of bi-angle tapes

There are several technologies for the realization of the thin plies, but tow-spread thinply technology is the most efficient and cost effective technology. In this approach, the tows are first pressed with the help of pneumatic system to spread the fiber bundles as uniform and closed to each other as possible without any damage to the fibers [START_REF] Sihn S | Experimental studies of thin-ply laminated composites[END_REF]. Figure 1-7 presents the tow spreading with pneumatic system while Figure 1-8 consists of a carbon fiber tow (CF12K) and its respective spread thin tows. When a polyester yarn is used for stitching having the sole purpose of consolidating the two dry plies only, then the stitching is known as non-structural stitching. However, there are NCFs in which glass fiber yarns are used instead of polyester yarns for enhancing the structural properties. Thus, this type of stitching is called as structural stitching [START_REF] Heb H | Elastic constants estimation of stitched NCF CFRP lamintes based on finite element unit-cell model[END_REF][START_REF] Carvelli V | Progressive damage in sticthed composites: Static tensile tests and tesnion-tension fatigue[END_REF].

Failure mechanisms in FRPs 1.5

In composites, failure is often an ill-defined term because being heterogeneous, FRPs typically exhibit many local failures prior to rupture into two distinct pieces, and hence 'first failure' does not necessarily show the 'final failure'. The local failures are thus suitably termed as 'damage' and the development of the local failure i.e damage with respect to increasing time or loading is termed as 'damage accumulation'. Damage refers to the loss of the load carrying capability of the material with the progressive damage growth. As a result, damage mechanics has been coined which describes the initiation and accumulation of the local damages (also known as diffused damages) dispersed throughout the continuum [START_REF] Herakovich | Mechanics of Fibrous Composites[END_REF]. In essence, damage mechanics signifies the effects of the micro-scale phenomena/damages on the macro-scale properties of the structures such as stiffness, strength, and fatigue life without the extensive discussion of the actual damage phenomenon occurring in the composite material in a phenomenological manner. To quote [START_REF] Dusan | Damage mechanics[END_REF] , "It is often argued that the ultimate task of the engineering research is to provide not so much a better insight into the examined phenomenon but to supply a rational predictive tool applicable in the industrial design". Due to the heterogeneity of the composite structures, a multi-scale approach is followed for the numerical study of the damage induced due to the external loads.

Multi-scale nature of the composite structure

Initiation of damage and propagation in composites is a multi-scale phenomenon and usually commences at the micro-scale in the matrix and fiber/matrix interface. Further increase of the load results to the growth of the matrix crack into meso-scale crack and delamination which finally transform into macro-cracks and ultimate failure. Normally, the damage models target either the micro-scale or meso-scale and predict the effects on the macro-scale properties of interest [START_REF] Camanho | Numerical Modelling of Failure in Advanced Composite Materials[END_REF]. Different scales for the damage modeling in continuous FRPs are explained with the help of Figure 12345678910.

The failure mechanisms or damage modes in FRPs are categorized in regards of these three different scales and are discussed in the following section.

Micro-level damage

The micro-level damage encompasses fiber breakage (Figure 1-12(a)), matrix microcracking and fiber/matrix de-bonding (Figure 1-12(b)) locally when the external loading surpasses the strength of the composite in that particular direction. Fiber breakage can be provoked by either tension or compression and usually higher amount of energy is dissipated when the fiber breakage takes place. In absence of any manufacturing defect in the matrix in terms of voids, fiber/matrix de-bonding is the weakest link to break by applying either tension or shear load. Hence, controlling the interfacial properties provide a way to control the performance of a composite structure.

Meso-level damage

Matrix-transverse cracking and splitting are the voids which are formed by the coalescence of the micro-damages: matrix micro-cracking, fiber/matrix de-bonding.

When matrix cracks are oriented perpendicular to the applied load then the matrix cracks are known as transverse cracks whereas when the matrix cracks are along the load then these are termed as splitting (van der Meer, 2012). Matrix-transverse cracking can be idealized as 3D tunneling and are propagated through the entire single ply of a laminate along the fiber direction either in width or length. Local delamination is also regarded as meso-damage which is initiated at the tips of the transverse cracks at the interface of the two neighbor plies. The meso-level damage modeling will be discussed in more detail in the last section of this chapter enabling to perform modeling based on this particular scale for UD FRPs which is one of the prime objectives of this thesis.

Macro-level damage

Separation of the two plies is termed as delamination, and it is considered as a macroscale damage. Delamination is deemed as a critical mode of failure due the fact that after delamination, the structure loses its load carrying capability. It can be provoked by either higher inter-laminar stress field, or in-plane loadings may also promote delamination. Notches, free edge effects and, or transverse matrix cracking in in-plane loading are some of the causes of delamination.

The damage modes described in the aforementioned section are shown in Figure 1-11, and Figure 1-12. It can be noticed in Figure 1-12 that the matrix-transverse cracking follows the fiber orientation in the particular ply. In multi-scale modeling, the scales can be linked with each other such as micro-scale unit cell can be linked with either meso-scale or macro-scale. Figure 1-13 outlines a typical implementation protocol of a multi-scale numerical model.

If a great level of detail is needed then the strains and stresses are computed at the constituent's level, and hence the constituents' configuration shall be provided: fiber shape, distribution, and material properties. This strategy is known as micro-scale. On the other side, the meso-scale for the UD FRPs is based on the ply-level, and the wellknown example is the elementary ply level elastoplastic damage coupled model developed by Ladevèze et al. [START_REF] Ladevèze P | Damage modeling of the elementary ply for laminated composites[END_REF]. The ply level modeling does not distinguish between the fiber and matrix rather a lamina is treated as a homogenized orthotropic continuum. The prime advantage of the multi-scale linking is that the material behavior of the upper scale can be computed from the lower scale known as multi-scale bottom-up approach. Recently, Garoz et al. developed a multi-scale model which computes the model parameters required in the Ladevèze model (total 13 materials properties and model parameters) by linking the micro-scale with the mesoscale and hence reduced the number of tests, the so called virtual micro-mechanical testing methodology [START_REF] Garoz | Material parameter identification of the elementary ply damage mesomodel using virtual micro-mechanical tests of a carbon fiber epoxy system[END_REF]. The fiber was treated as transversely isotropic with linear elastic constitutive law whereas epoxy was assumed as isotropic elastoplastic continuum. The fiber, matrix, and their interface strengths were provided in order to determine the model parameters. The damage in the fiber/matrix interface was modeled based on the cohesive law which is normally used for the delamination. There are also damage models exist as developed in [START_REF] Ulrich | Laminate damage model for composite structures[END_REF] , which are for the laminate level i.e structural level. The down side of the laminate level models is the stacking sequence dependency. The laminate level models are developed with the motivation to be used in the industrial applications by reducing the computation and modeling effort where layered shell elements are used for complex structure FEM modeling. A finer mesh is normally required in the ply-level energy based smeared crack damage models which are sometimes expensive in the industrial environment. The dependence of the ABAQUS built-in model developed by [START_REF] Lapczyk | Progressive damage modeling in fiber-reinforced materials[END_REF] on the element size is presented in Figure 1-14 [START_REF] Ulrich | Laminate damage model for composite structures[END_REF]. It can be comprehended that coarser the mesh faster is the damage accumulation and the corresponding softening. Hence, the ply level damage models can produce reliable results when finer mesh is used and, each layer is modeled with one element along the thickness that is not all the time feasible in the industry. Computationally efficient layered shell elements in larger composite structures are preferred. Even with coarser mesh, the numerical solution becomes ill-posed while evolving the damage. Keeping in mind to minimize the number of identification parameters required for a model, failure theories along with the fracture energies shall be used due to the fact that these parameters are readily available. The damage evolution shall be expressed in terms of fracture energy, critical strains and, or critical stresses in addition to a local characteristic dimensions that decreases the reliance on the finite element mesh.

Failure theories for FRPs 1.6

Failure theories, in principle, can be bifurcated into damage mode-independent and mode-dependent criteria. Orifici et al. [START_REF] Orifici Ac | Review of methodologies for composite material modelling incorporating failure[END_REF][START_REF] Daniel Im | Interfiber/interlaminar failure of composites under multi-axial states of stress[END_REF] presented the failure theories in a comprehensive manner in their articles which can be accurately adopted for the fiber failure, matrix failure, delamination initiation, and delamination propagation. There are about twenty failure theories in total which are available as a conclusion from the World Wide Failure Exercises (WWFEs) [START_REF] Hinton Mj | Failure criteria in fibre reinforced polymer composites: The World Wide Failure Exercise UK[END_REF]) (Soden et al., 1998b, Soden et al., 1998a[START_REF] Hinton | The background to Part B of the Second World-Wide Failure Exercise: Evaluation of theories for predicting failure in polymer composite laminates under threedimensional states of stress[END_REF]. It is pertinent to describe that no failure criterion is the winner but the accuracy varies from case to case and loading type. In most of the cases in Finite Elements Methods (FEM), the failure analysis is based on some failure criterion which accepts the stresses and strains for each lamina in the local coordinate system/material coordinate system. And for this purpose, orientation of each ply, thickness and elastic properties must be provided as inputs to the FEM program. Most of the failure theories have been developed for the UD orthotropic lamina such as Hashin theory [START_REF] Hashin | Failure criteria for unidirectional fiber composites[END_REF]) and Puck's theory. However, Hashin-Rotem criteria has been employed for the woven plies in [START_REF] Daniel | Three-dimensional characterization of textile composites[END_REF][START_REF] Liu | Experimental and numerical studies on the impact response of damage-tolerant hybrid unidirectional/woven carbon-fibre reinforced composite laminates[END_REF]. In the present work, Puck's theory (Puck, 2002a, Puck, 2002b, Puck et al., 2002) has been used for triggering the intra-laminar damage in UD FRPs due to some of its attractive features highlighted in the sequel.

Generally, most of the available failure theories are based on the 'best fit' of the experimental data which are transformed into mathematical expressions commonly in quadratic forms, but there are some theories which are based on the physical reasoning of the composite failure mechanisms. For the structure designer point of view, a very common approach is to calculate the failure index for a 3D state of applied stress for each individual lamina based on the variety of failure theories. Polynomial-fit criterion, for instance, Tsai-Wu [START_REF] Tsai | A General Theory of Strength for Anisotropic Materials[END_REF]) is commonly used in the industry due to its simplicity but it does not distinguish the fiber and matrix failure mode. However, there are more advanced failure criteria exist such as Puck's failure theory (Puck, 2002a, Puck, 2002b, Puck et al., 2002). It is based on the physically observable damage. It predicts not only the damage onset, but the failure mode along with the failure planes.

Puck's failure theory is of higher interest as far as failure is concerned due to the reason that it enables us to foresee the kind and the location of a fracture which do matter [START_REF] Matthias Deuschle | Finite element implementation of Puck's failure theory for fibre-reinforced composites under three-dimensional stress[END_REF]. Failure criterion shall be simpler enough to be used in the engineering and industrial applications but in parallel, it shall be equipped to capture effectively the corresponding physics of the failure phenomenon (Puck et al., 2002, Puck, 2002b).

Hashin's failure theory [START_REF] Hashin | Failure criteria for unidirectional fiber composites[END_REF] is widely used in the models though it may not predict the matrix damage with high precision as highlighted in [START_REF] Lapczyk | Progressive damage modeling in fiber-reinforced materials[END_REF]. Camanho and Davilla (Camanho, 2003) which was originally for the inter-laminar damage: cohesive damage accumulation for modeling delamination. The model proposed by Lapczyk et al. is available in the commercial FEM software ABAQUS using Hashin failure criterion [START_REF] Hashin | Failure criteria for unidirectional fiber composites[END_REF], and hence called Hashin model in the ABAQUS material library. In the present research work, Puck's theory will be employed in the same context as the Hashin's theory has been used in [START_REF] Chen | A combined elastoplastic damage model for progressive failure analysis of composite materials and structures[END_REF][START_REF] Vasiukov | Non-linear material modeling of fiber-reinforced polymers based on coupled viscoelasticity-viscoplasticity with anisotropic continuous damage mechanics[END_REF]. The accuracy of the matrix meso-damage initiation of Puck's theory has been demonstrated in [START_REF] Lee | Initial and progressive failure analyses for composite laminates using Puck failure criterion and damage-coupled finite element method[END_REF] in which the authors predicted the matrix damage with 95.8% accuracy for different combined experimental load cases. Similar conclusions were also made in number of scientific studies [START_REF] Perillo | Numerical application of threedimensional failure criteria for laminated composite materials[END_REF][START_REF] Philippidis | A progressive damage FEA model for glass/epoxy shell structures[END_REF]. [START_REF] Perillo | Numerical application of threedimensional failure criteria for laminated composite materials[END_REF] have predicted the ultimate breaking load capacity of Open Hole Tension (OHT) composite coupon having relative less error with the experiments by using Puck's theory as compared to Hashin's theory. It was recommended by the authors [START_REF] Perillo | Numerical application of threedimensional failure criteria for laminated composite materials[END_REF]) that Puck's theory shall be used in conjunction with the gradual damage evolution rather than sudden degradation in order to minimize the deviation from the experimental values. Hence, Puck's theory seems to be a better candidate to be used for the damage initiation based on the prediction accuracy with the elastoplastic constitutive law.

Lapczyk et al. generalized the damage growth law of

Puck's theory has been used earlier with the progressive damage evolution by number of researchers. In fact this theory has its own damage degradation as well [START_REF] Knops | Analysis of Failure in Fiber Polymer Laminates: The Theory of Alfred Puck[END_REF][START_REF] Knops | Gradual failure in fibre/polymer laminates[END_REF]. However, the range of the parameters is very broad and the secant elastic properties are used rather than the initial elastic properties of the lamina in order to capture the pre-Inter Fiber Fracture (IFF) non-linearity caused by the plasticity and micro-damage in the matrix material. In Puck's theory, IFF signifies the matrix fracture in a lamina across the whole thickness of that particular lamina which takes place by the coalescence of diffused damages that include matrix micro-cracking, fiber/matrix debonding referred by point B in Figure 1-16-a. Different types of IFFs and the corresponding stresses are elaborated in Figure 1-15. The transverse compressive stress and the out-of-plane shear stress produce inclined fracture.

Zhong et al. [START_REF] Zhong | A continuum damage model for threedimensional woven composites and finite element implementation[END_REF] proposed the damage evolution in woven composites by considering the 3D Puck's theory for the damage onset in the fiber yarn because of the capability to predict the damage as well as the orientation of the damage. In addition, [START_REF] Lee | Initial and progressive failure analyses for composite laminates using Puck failure criterion and damage-coupled finite element method[END_REF] used the plane stress case of Puck's theory in CFRP OHT where an instantaneous unloading of the corresponding finite element has been accomplished, a sudden death concept, when one of any five Puck's failure indices turns into unity in that particular finite element. A series of publications has been presented by Schuecker et al. [START_REF] Schuecker | A continuum damage model for fiber reinforced laminates based on ply failure mechanisms[END_REF], Schuecker, 2008a, Schuecker, 2008b) in which the physically based Puck's theory has been efficiently exploited to its best to develop a robust CDM model. The model [START_REF] Schuecker | A continuum damage model for fiber reinforced laminates based on ply failure mechanisms[END_REF], Schuecker, 2008a, Schuecker, 2008b) stems on the Puck's theory and the determination of the damaged compliance tensor on Mori-Tanaka method [START_REF] Mori | Average stress in the matrix and average elastic energy of materials with misfitting inclusions[END_REF]. The crack induced by the external loading was termed as inclusion whose shapes were assumed as oblated spheroidized, having a known aspect ratio as a priori, and then the Eshelby tensor for each inclusion was determined in the fracture plane coordinate system using Puck's theory in a transversely isotropic host continuum. Knowing the Eshelby tensor, the damage compliance tensor for a ply was computed. This model [START_REF] Schuecker | A continuum damage model for fiber reinforced laminates based on ply failure mechanisms[END_REF], Schuecker, 2008a, Schuecker, 2008b) is attributed with some aided unique features over the other models as it defines a sub-surface inside the Puck's envelope which captures the initiation of the micro-damages (point A in Figure 1-16-a), the predecessors of the meso-damage or in other words the IFFs, as shown in Figure 1-16-a (Schuecker, 2008b) and Figure 1-16-b [START_REF] Knops | Gradual failure in fibre/polymer laminates[END_REF]. . This is the reason why the transverse compression was further categorized in moderate and higher values of transverse compression in comparison with the inplane shear stress by Puck et al. (Puck et al., 2002, Puck, 2002b). These are called as Mode-B and Mode-C in 2D version of Puck's theory. Therefore, total five equations are available for the full description of the 2D Puck's theory as employed in [START_REF] Lee Cs | Initial and progressive failure analysis of composite laminates using Puck failure criterion and damage-coupled finite element methode[END_REF]. In Mode-C the fracture plane angle fp θ is oblique and is in the range 24°-54° (see, -17). To compute fp θ in Mode-C in plane stress case, an analytical form is available. Due to the wedge shape oblique cracks induced in Mode-C, this mode is regarded very harmful to the elastic properties of the lamina in comparison to the cracks in Mode-A and Mode-B [START_REF] Hinton Mj | Failure criteria in fibre reinforced polymer composites: The World Wide Failure Exercise UK[END_REF]. The failure plane of a 90°off-axis laminate in compression test of CFRPs is shown in Figure 1-19 [START_REF] Wang | Modeling of nonlinear response in loading-unloading tests for fibrous composites under tension and compression[END_REF]. The fracture angle in CFRPs is normally 53°± 2°. The inclination parameter in compression can also be determined with the help of this test knowing the fracture angle (Puck et al., 2002, Puck, 2002b).

Damage modeling in FRPs 1.7

The accurate though easier reliable modeling of mechanical behavior of Fiber Reinforced Polymers (FRPs) composite structures is a challenging and arduous task to accomplish in the today's stringent industrial requirements that strive persistently to seek cost-effective solutions. The contributing factors in the complexity are the heterogeneity, anisotropy, and the multi-scale nature of the composite structures which usually need many characterization tests for the qualification of the structures to be introduced into their intended service conditions. The number of the certification tests is large, as for example, a typical large composite airframe need 10,000 tests in order to pass the aerospace safety standard (van der Meer, 2012). To minimize these testing in the development and to squeeze the qualification timeline of the project and, hence cost, reliable numerical tools shall be synergized with the characterization plan in order to compete in the market. For the efficient use of the composite structures up to the ultimate failure, complete failure analysis shall be carried out including the damage initiation and saturation till the final rupture [START_REF] Liu | Recent developments on damage modeling and finite element analysis for composite laminates: A review[END_REF]. The conventional design approach which declares a laminate to be failed or safe is based on the First Ply Failure (FPF) criterion and is very conservative (Schuecker, 2008b). So, failure analysis has to be progressive instead of abrupt unloading of the structure. To address this issue, the progressive or continuum damage mechanics analysis shall be considered where the damage is continuously evolving as a function of the external load.

Progressive damage modeling

The catastrophic failure of a composite structural element (composite panels, wind turbine blade, etc.) rarely takes place at the load level corresponding to the initial ply failure. The structures based on FPF approach are usually over-designed which augments the weight of the structures which is one of the critical design parameters in transportation: aerospace, automotive, and so on. Progressive damage modeling is based on the theory of Continuum Damage Mechanics (CDM). In CDM the cracks are smeared-out or homogenized over the continuum and the degraded stiffness is used in the stress-strain relations (constitutive law) to predict the response of the damaged continuum. It initiated with the work of Kachanov for the creep analysis of isotropic materials [START_REF] Kachanov | Time of the rupture process under creep condition[END_REF], and has been remained an indispensable tool for the damage analysis of the structures. The application of the CDM approach has multiplied rapidly with the passage of time. It is based on the irreversible thermodynamic processes having internal state variables which can be expressed by a family of scalars or in tensorial notation [START_REF] Chaboche | Continuum damage mechanics: Part 1-General concepts[END_REF][START_REF] Chaboche | Continuous damage mechanics -A tool to describe phenomena before crack initiation[END_REF][START_REF] Maire | A new formulation of continuum damage mechanics for composite materials[END_REF].

The choice of the state variables depends on the material damage modes, and it stores the history that what happened to the material throughout its loading range. It is advised to use scalar damage variables where the micro-structure of the materials limits the directions of the damage. For instance in FRPs, the stiffer reinforcement defines the direction of the cracks [START_REF] Maire Jf | A new formulation of continuum damage mechanics for composite materials[END_REF]. This reduces the rank of the damage tensor and simplifies the model. The application of CDM to a multi-dimensional damage problem consists of three major steps to be followed [START_REF] Maire Jf | A new formulation of continuum damage mechanics for composite materials[END_REF]:

1. The first step is to define the damage variables as state variables along with the effective stress σ  . Effective stress is defined as the stress which is transferred by the fictitious undamaged material. The domain is assumed as a continuum where the field equations are smooth and continuous regardless of the presence of micro-damage and cracks formed by the applied load. For composite materials reinforced with stiff and strong fibers, damage can be accurately represented by a second-order damage effect tensor ( )

d 
in which the principal directions are aligned with the material direction. The effective stress vector σ  is related to the actual Cauchy stress vector σ by the damage effect tensor ( )

d 
which is the function of the scalar damage variables as:

( ) . d = σ σ   (1.1)
The actual Cauchy stress is known as nominal stress in CDM which is the stress transferred by the damaged material. In FEM program, this nominal stress shall be updated and compared with the experimental results because this is the actual stress which is transferred by the damaged material.

2. The second phase is to formulate the state laws (relation between the internal state variables and the associated conjugate independent variables. This is accomplished by employing the thermodynamic potentials for anisotropic composite materials. The scalar values thermodynamic potentials represent the state of a system which depends upon the internal state variables and the corresponding observable conjugate variables (thermodynamic driving forces).

If the internal state variable is a scalar then the corresponding driving force will also be scalar, and so on. There are two thermodynamic potentials which can be represented in quadratic form as a function of the observable variables and internal state variables denoted by Λ [START_REF] Maire Jf | A new formulation of continuum damage mechanics for composite materials[END_REF]:

1 ( , : : (Helmholtz) 2 1 ( , : : (Gibbs) 2 Ψ = Φ = ε Λ) ε C ε σ Λ) σ S σ   (1.2)
Here, C  and S  are the effective stiffness and compliance tensors. The state laws are hence formulated as:

:

:

∂Ψ = = ∂ ∂Φ = = ∂ σ C ε ε ε S σ ε   (1.3)
The damage stiffness and compliance tensors C  and S  can also be derived by knowing the expression of either of the thermodynamic potentials:

2 2 2 2 ∂ Ψ = ∂ ∂ Φ = ∂ C ε S σ   (1.4)
The damage effect tensor ( )

d 
is also derived after having the pristine materials stiffness tensor and the damaged stiffness tensor exploiting the strainequivalence or energy-equivalence assumption. Taking the strain-equivalence assumption ( ) d  takes the following form [START_REF] Chaboche | Continuous damage mechanics -A tool to describe phenomena before crack initiation[END_REF]

: 1 ( ) : d - = C C   (1.5)
The choice to use one of the potentials depends upon the model developer.

Model which is developed based on Gibbs potential ( , Φ σ Λ) will be in terms of the stress which can be easier to identify experientially in contrary to the model employing ( , Ψ ε Λ) which is in terms of strain tensor. On the FEM implementation side, strain driven models are relatively easier to embed in a code. Implementation is done once for a new model but the model identification is a recurrence task which has to be performed if the grade of the materials is altered. Consequently, Gibbs potential ( , Φ σ Λ) is widely used.

3. In the final step of developing a model in the framework of thermodynamics, the kinetic laws governing the rate of damage and hardening are formulated which are functions of the damage and hardening potentials [START_REF] Maire Jf | A new formulation of continuum damage mechanics for composite materials[END_REF].

Here, the second law of thermodynamics shall be satisfied for irreversible processes (damage and, or plasticity). For solid materials, it reduces to Clausius-Duhem in-equality and is also called the dissipation inequality. Using the Gibbs potential ( , Φ σ Λ) , the energy dissipated per unit volume by the irreversibility of all the internal variables ( ) i Λ involved is rendered as:

. 0 . 0

n i i i n i i ∂Φ Ξ = ≥ ∂ Ξ = ≥ ∑ ∑ i Λ Λ Y Λ   (1.6) i Y
is the conjugate thermodynamic driving force which may be scalar or tensor depending on its internal variable. As i Y is based on the Gibbs potential ( ,

Φ σ Λ)
which ensures the thermodynamic consistency. Hence, the evolution of the internal variables must be monotonically evolved in order to satisfy positive dissipation.

Damage mechanics and phenomenological approach

It would be computationally inefficient to study and incorporate the physical intralaminar damages in the model for the structural scale. So alternatively, the damages are model in a phenomenological manner and the models based on this philosophy are known as phenomenological models. Such models are based on the mathematical theory of an observed phenomenon without paying attention to the actual physics of the process taking place during the rupture of the materials. Being a non-destructive quantity, stiffness degradation is preferred to measure experimentally than the strength as the damage growth takes place. Moreover, stiffness degradation is more sensitive to the load, static or fatigue, in contrast to strength. Examples of the phenomenological models can be found in (W. [START_REF] Van Paepegem | Modelling damage and permanent strain in fibrereinforced composites under in-plane fatigue loading[END_REF][START_REF] Gagel A | On the damage evolution and the mechanical dgradation of Glassfiber non-crimp fabrics reinforced epoxy under tesnile loading: Experiment and finite element modeling[END_REF][START_REF] Ladevèze P | Damage modeling of the elementary ply for laminated composites[END_REF][START_REF] Schuecker | A continuum damage model for fiber reinforced laminates based on ply failure mechanisms[END_REF], etc. Vast majority of the phenomenological models are based on CDM, and are defined in the framework of the theory of irreversible thermodynamics.

This approach enables us to predict the residual/degraded properties as a function of the applied load.

It was pointed out in [START_REF] Xavier | New advances in damage mechanics and computational methods for composites: from research to industry for spatial applications[END_REF]) that there exist a considerable hurdle in implementing the models which are developed in the research labs/universities, and transform those models into useful work for the industrial sectors due the complication and parameters identification. Hence, following are the key features a researcher shall keep in mind while devising a prediction model (Ribeiro L M, 2012):

• must have higher accuracy

• be simple to be implemented and possesses low computational cost

• require simple characterization tests for model validation and identification

• need simple composite coupons to be manufactured

Meso-scale damage modeling

Damage modeling at the meso-scale is defined based on the type and architectural topology of the FRPs. If a laminate is embedded with only UD plies then the ply-level is considered as the meso-scale as the elementary ply level model developed in [START_REF] Ladevèze P | Damage modeling of the elementary ply for laminated composites[END_REF]. There are numerous other meso-scale models available for FRPs, few of them include (Maimí et al., 2007a, Chen et al., 2012[START_REF] Vasiukov | Non-linear material modeling of fiber-reinforced polymers based on coupled viscoelasticity-viscoplasticity with anisotropic continuous damage mechanics[END_REF][START_REF] Boutaous A | A new modeling of plasticity coupled with damage and identification for carbon fiber composite laminates[END_REF][START_REF] Reinoso | A consistent anisotropic damage model for laminated fiber-reinforced composites using the 3D-version of the Puck failure criterion[END_REF]. Indeed, the ply-level damage modeling results the model as a stacking-independent model which imparts versatility to the model in regards of the stacking sequence.

For other topologies of the fabrics like textile or NCFs fabrics, the meso-scale is different. The main heterogeneities in the NCFs based laminate on the meso-scale are the fiber bundles (tows/yarns), in-plane waviness of the fibers, and the resin-rich zones caused by the through-thickness stitching in a single lamina. Therefore, these mentioned heterogeneities shall be incorporated in the modeling either by selecting an appropriate Representative Volume Element (RVE), or by applying some homogenization method to get the effective properties of a lamina [START_REF] Marklund E | Stiffness and strength modeling of non-crimp fabric composites. 52nd Structures[END_REF]. In addition, the individual fibers are not modeled as in the case of micro-scale analysis, but the yarns are assumed as the homogenous solid material just like the UD ply for which the transversely isotropic assumption is valid (Gonzales A, 2010). To establish a simplified meso-model for NCFs, the following assumptions are viable to consider [START_REF] Lomov | Non-crimp fabric composites : Manufacturing, properties and applications[END_REF].

1. The fine polyester yarn can be excluded in the damage mechanics analysis because it has nearly the same stiffness contributions as of matrix. Nevertheless, the disturbance caused by the stitching yarn shall be taken into consideration to appraise the influence on the damage onset and evolution.

2. Plies are modeled straight, hence out-of-plane crimp if any, is ignored.

3. Due to the sophisticated technology of tow-spreading, continuous fiber plies shall be modeled. The local elastic properties of the fibrous regions are computed analytically with the help of the formulas of Chamis [START_REF] Chamis | Mechanics of Composite Materials: Past, Present, and Future[END_REF].

Plasticity and damage coupling in FRPs 1.8

In principle, the non-linear material behavior is characterized by two different mechanical phenomena regardless the material is metal or composite as reported in [START_REF] Abu | On the coupling of anisotropic damage and plasticity models for ductile materials[END_REF]. The first source of non-linearity is the plasticity which in case of metals is the slippage of the dislocations along the slip planes, whereas in composites is the irreversible deformation in the softer matrix material. The second cause of the nonlinearity is the possible accrued damages which encompass all the micro-voids/cracks, fiber/matrix de-cohesion, meso-cracks, etc. Upon the damage accumulation and growth by the exerting load, the drop in the elastic modulus and also in the plastic hardening is governed by applying the coupled constitutive equations of damage and plasticity [START_REF] Lemaitre | Coupled elasto-plasticity and damage constitutive equations[END_REF]. In the coupled damage/plasticity CDM models, it is assumed that the drop in the slope (stiffness degradation) of the stress-strain curve is partially associated to the plasticity and partly to the damage induced (O 'Higgins et al., 2011, O'Higgins et al., 2012b, O'Higgins et al., 2012a). It is also observed in some FRPs that plastic deformation take place prior to the damage initiation, so in this case there is only plasticity without stiffness degradation. Beyond the damage initiation both the plasticity and stiffness degradation exist simultaneously [START_REF] Chen Jf | A combined elastoplastic damage model for progressive failure analysis of composite materials and structures[END_REF]. The damage and plasticity ultimately induce inelastic strains which can be additively decomposed into elastic strain vector e ε , plastic strain vector p ε , and inelastic strain vector due to damage d ε [START_REF] Abu | On the coupling of anisotropic damage and plasticity models for ductile materials[END_REF]. The total strain vector ε can be read as:

= + + e p d ε ε ε ε (1.7)

Plasticity and damage potentials

In order to specify the plasticity and damage thresholds points, the plastic and damage potentials are necessary to define. There are two main types of the yield functions that can be applied for non-linear material modeling of FRPs: (i) hydrostatic stress dependent, (ii) hydrostatic stress independent. The hardening law may be based on either kinematic, and or isotropic. The combination of both formulated in a single plasticity potential may give better results which is called as combined hardening but, generally, if the loading is monotonic then isotropic hardening is preferred and when there is the load reversal or cycling, then kinematic hardening may predict the nonlinearity with better accuracy than the isotropic [START_REF] Dunne F | Introduction to computational plasticity[END_REF]. Damage is also modeled in a similar fashion as of plasticity by defining the damage surface and the corresponding damage hardening law. In general form, the plasticity surface denoted as f and the damage surface denoted as g are expressed as:

( ) σ ( ) 1 0 eq f = - = σ,d, X σ,d, X   (1.8) ( , ) ( , ) 1 0 eq g Y = - = Y H Y H (1.9)
Here in eq.(1.8), σ  is the effective stress tensor, d is the damage tensor, and X is the back stress tensor used in the kinematic hardening. In eq.(1.9), Y denotes the thermodynamic force tensor that drive the damage, and H expresses the thermodynamic back stress tensor analogous to X . The plasticity and the damage surfaces delimit the plasticity and the damage in the stress space σ  and in the thermodynamic space Y ,respectively. eq σ and eq

Y are the scalar equivalent quantities and it depends on the type of potential used. For example, the model proposed in [START_REF] Chen | Simulating progressive failure of composite laminates including in-ply and delamination damage effects[END_REF] it is expressed as ( )

2 2 22 12 3 σ 2 2 eq a σ σ = +   . Conditions ( ) 0 f < σ,d, X
, and ( , ) 0

g < Y H
signify that the loading step is elastic and there is no damage evolution. If both conditions are equal to zero then it indicates that plasticity and damage both shall be accumulated. Conditions (

) 0 f > σ,d, X and ( , ) 0 g > Y H
are inadmissible. Dissipation function is expressed as the sum of the product of the thermodynamic forces and the internal variables as:

: : 0 = - - - ≥ σ ε Y : d X H     P a p G :
(1.10)

Using associative flow rules ( and

) f F G g = =
, the relation between the thermodynamic driving force variables and their conjugate internal variables can be established:

p d d p d F G G F G λ λ λ λ λ ∂ ∂ = ∂ ∂ ∂ = ∂ ∂ = - ∂ ∂ = ∂ ε σ σ d Y X H          + a p G (1.11)
Flow rules and Kuhn-Tucker consistency conditions are used for the determination of the Lagrange plastic and damage multipliers denoted by p λ  and d λ  , respectively. 0 ; 0 ; 0 0 ; 0 ; 0

p p d d f f g g λ λ λ λ ≥ ≤ = ≥ ≤ =     (1.12)
Most of the models based on the CDM coupled with plasticity do not consider the inelastic strains caused by the irreversible micro-damages as they assume that all the inelastic strain is due to the plasticity of the matrix in the composites. For instance, the phenomenological elastoplastic damage model proposed by Ladevèze et al. [START_REF] Ladevèze P | Damage modeling of the elementary ply for laminated composites[END_REF] is very efficient in the prediction of the transverse tensile and shear non-linear mechanical behavior of FRPs. Though this model portrays the complete non-linearity caused by the combined effect of the micro-damages and plasticity with significant accuracy from the very beginning depending on the thermodynamic force thresholds, as is generally the actual case in composites, it demands some non-conventional testing. A series of publications (O' Higgins et al., 2011, O'Higgins et al., 2012b, O'Higgins et al., 2012a[START_REF] Malgioglio | Material parameter identification challenge and procedure for intra-laminar damage prediction in unidirectional CFRP[END_REF] are available which contain very comprehensive elaboration of the model parameters identification of this particular model [START_REF] Ladevèze P | Damage modeling of the elementary ply for laminated composites[END_REF]. In the response to minimize the unconventional testing for model identification, there are models [START_REF] Chen | A combined elastoplastic damage model for progressive failure analysis of composite materials and structures[END_REF], Maimí et al., 2007a[START_REF] Vasiukov | Non-linear material modeling of fiber-reinforced polymers based on coupled viscoelasticity-viscoplasticity with anisotropic continuous damage mechanics[END_REF][START_REF] Reinoso | A consistent anisotropic damage model for laminated fiber-reinforced composites using the 3D-version of the Puck failure criterion[END_REF][START_REF] Schuecker | A continuum damage model for fiber reinforced laminates based on ply failure mechanisms[END_REF][START_REF] Lapczyk | Progressive damage modeling in fiber-reinforced materials[END_REF][START_REF] Vaziri | Finite element analysis of fibrous composite structures: A plasticity model[END_REF] which make use of the readily available material properties both for damage initiation and evolution. Strength parameters are used for the damage onset whereas the fracture energy is used in the damage accumulation formulation in such models, in contrary to the model of Ladevèze et al. [START_REF] Ladevèze P | Damage modeling of the elementary ply for laminated composites[END_REF]. As composites exhibit higher degree of non-linearity specifically in case of shear when the fibers are oriented along ±45°. Therefore, in such circumstances it becomes necessary to employ non-linear constitutive law. Nonlinearity can be either incorporated using classical plasticity approach. Or alternatively, a higher order polynomial can be used which is obtained by fitting the experimental shear stress-strain data. The approach of using polynomials not only simplifies the formulation of the model considerably, but also saves time while running the FEM simulations. Such non-linearity was introduced in (Van Paepegem et al., 2006a, Van Paepegem et al., 2006b), and is widely used in the damage models. Some examples are referred in (Schuecker, 2008a, Falzon, 2011a, Falzon, 2011b). These models consider the non-linearity in shear stress only in FRPs.

Return Mapping Algorithm (RMA)

Return Mapping Algorithm (RMA) is broadly used in the plasticity material modeling [START_REF] Simo | Consistent tangent oprators for rate-independent elastoplasticity[END_REF]. RMA transforms the problem into finding the closest distance of a point (trial point) to a convex set of points defined by the plasticity potential in the given problem. The geometrical interpretation of RMA is illustrated in Figure 1-20. This procedure will be followed in numerical modeling in the next chapter. Following are the two basic steps involved in the RMA:

Figure 1-20: Geometrical representation of Return Mapping Algorithim (RMA) [START_REF] Simo | Consistent tangent oprators for rate-independent elastoplasticity[END_REF] Step 1: Elastic trial state: Based on the elastic constitutive law assumption, the trial stress is computed. The trial stress is plugged-in in the plastic potential to evaluate that whether the step is elastic or plastic.

Step 2: Plastic Corrector: In case of non-linear load step diagnosed in step 1, the system of equation is solved by enforcing the flow rules and Kuhn-Tucker conditions.

Such conditions curtail the problem to find the root of a non-linear expression having only one unknown. This unknown is non-negative number and is called Lagrange multiplier/factor which is computed with the iterative procedure of Newton-Raphson scheme.

Conclusion

WWFEs and other studies discussed in this chapter have shown that Puck's criterion is one of the best criteria to predict the failure of FRPs laminate composites. Recent articles reveal that this criterion can also be used as a good indicator to detect the mesodamage initiation. Furthermore, crack orientation and the degree of severity of the fracture can also be distinguished due to the physical background of Puck's theory, and still using the same number of strength parameters that are required for other conventional failure theories. Total five strength parameters are needed for Puck's theory. Therefore, Puck's theory is a better candidate to be used in the framework of CDM. In the light of these arguments, a damage model is developed in the next chapter based on the Puck's theory for the plane stress case to provide insights into the damage mechanisms in the continuous FRPs composite materials.

Introduction 2.1

This chapter pertains to the detailed discussion of the anisotropic damage modeling on the basis of CDM theory coupled with plasticity. The thermodynamically consistent damage evolution law is defined to accumulate the damage leading to the subsequent stiffness degradation coupled with the isotropic hardening plasticity. The model uses one parameter isotropic hardening function. Due to the fact that the initial yield strength value cannot be determined very accurately in the experiments of FRPs, therefore, this plasticity function does not use any initial yield strength. In case of no initial yield strength, plasticity is initiated from the first increment of the applied load in the model. However, the amount of plasticity is trivial and does not affect the stress-strain response which can be comprehended after the comparison of the predicted response with the experimental counterpart. Finite Element Method (FEM) is employed to predict the mechanical behavior at quasi-static loading condition. The damage initiation and the evolution are formulated keeping in view the 2D version of the Puck's failure theory.

The model relies on the strength values required for a ply and the fracture energies in addition to the plasticity potential parameters.

Damage/plasticity coupled material modeling 2.2

Following [START_REF] Maire Jf | A new formulation of continuum damage mechanics for composite materials[END_REF], three steps have to be used to develop a damage model in the framework of CDM as discussed in great detail in chapter 1.

• Definition of the damage variables

• Definition of the state laws

• Definition of the kinetic laws governing the rate of damage and hardening Some of the meso-models that are based on the coupling of plasticity and damage are discussed briefly in the following sub-section.

Damage/plasticity coupled meso-models

The model developed by [START_REF] Ladevèze | Damage modelling of the elementary ply for laminated composites[END_REF] is one of the important models in the composite community that utilized this coupling. Similarly, the progressive failure models developed by Vaziri et al. [START_REF] Vaziri | Finite element analysis of fibrous composite structures: A plasticity model[END_REF], Chen et al. [START_REF] Chen | A combined elastoplastic damage model for progressive failure analysis of composite materials and structures[END_REF][START_REF] Chen | Simulating progressive failure of composite laminates including in-ply and delamination damage effects[END_REF], and Vasiukov et al. [START_REF] Vasiukov | Non-linear material modeling of fiber-reinforced polymers based on coupled viscoelasticity-viscoplasticity with anisotropic continuous damage mechanics[END_REF] are some of the elastoplastic/damage models that aimed the meso-scale damage modeling of UD plies based FRPs laminates. Chen et al. [START_REF] Chen | A combined elastoplastic damage model for progressive failure analysis of composite materials and structures[END_REF]) selected one parameter plastic potential for plane stress case with isotropic hardening function that was developed in [START_REF] Sun | A simple flow rule for characterizing non-linear Behavior of Fiber composites[END_REF], and was used in [START_REF] Weeks | Modeling the non-linear rate-dependent behavior in fiber-reinforced composites[END_REF]. Vasiukov et al. [START_REF] Vasiukov | Non-linear material modeling of fiber-reinforced polymers based on coupled viscoelasticity-viscoplasticity with anisotropic continuous damage mechanics[END_REF], used the modified Hoffman criterion having hydrostatic pressure sensitivity and non-linear kinematic hardening in 3D stress space. When the plasticity potentials are used for composites, plasticity flow only develops in matrix and is blocked in the direction of fibers. In the above models [START_REF] Chen | A combined elastoplastic damage model for progressive failure analysis of composite materials and structures[END_REF][START_REF] Chen | Simulating progressive failure of composite laminates including in-ply and delamination damage effects[END_REF][START_REF] Vasiukov | Non-linear material modeling of fiber-reinforced polymers based on coupled viscoelasticity-viscoplasticity with anisotropic continuous damage mechanics[END_REF], Hashin's failure theory [START_REF] Hashin | Failure criteria for unidirectional fiber composites[END_REF] has been used for the damage initiation like in most of the other models: for instance in [START_REF] Lapczyk | Progressive damage modeling in fiber-reinforced materials[END_REF][START_REF] Jingran | A coupled elastic-plastic damage model for the mechanical behavior of three-dimensional (3D) braided composites[END_REF]. Improved ultimate breaking strength along with the matrix nonlinearity for OHT coupon of T300/1034-C Carbon epoxy were reported in [START_REF] Chen | A combined elastoplastic damage model for progressive failure analysis of composite materials and structures[END_REF][START_REF] Vasiukov | Non-linear material modeling of fiber-reinforced polymers based on coupled viscoelasticity-viscoplasticity with anisotropic continuous damage mechanics[END_REF], in comparison to the previous models (Maimí et al., 2007a, Maimí et al., 2007b[START_REF] Chang | A Progressive Damage Model for Laminated Composites Containing Stress Concentrations[END_REF][START_REF] Sc | A Progressive Failure Model for Composite Laminates Containing Openings[END_REF]. In addition, Maimí et al. (Maimí et al., 2007a, Maimí et al., 2007b) utilized the LaRC03 criterion and employed in-situ ply strength in the damage activation functions. The in-situ strength interprets the effects of the lamina thickness and its position in the laminate. The approach of Maimí et al. (Maimí et al., 2007a, Maimí et al., 2007b) also considers a fixed oblique fracture plane in transverse compression instead of addressing the angle as a continuous variable. These models which are based on the failure theories usually pose the convergence problems, and the solution aborts prematurely once the FF or IFF is diagnosed. It can be addressed by incorporating appropriate artificial viscous effects in post-failure loading phase by introducing the viscous regularization coefficient usually denoted byη [START_REF] Chen | A combined elastoplastic damage model for progressive failure analysis of composite materials and structures[END_REF][START_REF] Lapczyk | Progressive damage modeling in fiber-reinforced materials[END_REF][START_REF] Vasiukov | Non-linear material modeling of fiber-reinforced polymers based on coupled viscoelasticity-viscoplasticity with anisotropic continuous damage mechanics[END_REF], Maimí et al., 2007a, Maimí et al., 2007b). The influence of the viscous regularization coefficient has been highlighted in [START_REF] Lapczyk | Progressive damage modeling in fiber-reinforced materials[END_REF], therefore, it must be selected carefully so that the results are not significantly affected.

Anisotropic plasticity damage coupled constitutive law

The effective stress vector σ  and the actual Cauchy stress vector σ are linked via the damage effect tensor ( )

d 
which is the function of the scalar damage variables as:

( ) . d = σ σ   (2.1)
Here for plane stress assumption the damage effect tensor is expressed as: (1 )(1 ) and2 2 2 (1 )(1 )

1 2 21 1 1 1 ( ) ; ; 1 1 1 d diag d d d   =   - - -   
T C d d d = - -
T C d d d = - - .

Meso-damage onset functions

The commonly used failure criteria predict the meso-damage which is the consequence of the micro-damages: matrix micro-cracking, de-bonding, etc. As a result, the nonlinearity caused by the micro-damage prior to the meso-damage is overlooked in this manner, and it is assumed to be attributed solely to the plasticity of the matrix. One possible approach exists to define a sub-failure surface interpreting the initiation of the micro-damage/diffused damage inside the meso-damage failure surface. Böhm et al. [START_REF] Böhm | A phenomenologically based damage model for 2D and 3D-textile composites with non-crimp reinforcement[END_REF] applied Cuntze failure theory [START_REF] Cuntze | Efficient 3D and 2D failure conditions for UD laminae and their application within the verification of the laminate design[END_REF] with the sub-failure surface and, Scheucker et al. [START_REF] Schuecker | A continuum damage model for fiber reinforced laminates based on ply failure mechanisms[END_REF], Schuecker, 2008a, Schuecker, 2008b) employed the Puck's failure theory (Puck et al., 2002, Puck, 2002b): sub-failure surface defines the diffused damage threshold whilst the outer surface defines the loci of the meso-damage initiation. The model [START_REF] Schuecker | A continuum damage model for fiber reinforced laminates based on ply failure mechanisms[END_REF], Schuecker, 2008a, Schuecker, 2008b) exhibited its capability in uni-axial as well as multi-axial loading along with the load reversals, but the implementation possesses the difficulty. Nevertheless, this sub-surface inclusion in the model increases the requirement of the additional diffused damage initiation thresholds to be determined experimentally for the material modeling.

More recently, the 3D form of Puck's theory has been used for triggering the intralaminar damage by coupling the elastic constitutive law with CDM [START_REF] Reinoso | A consistent anisotropic damage model for laminated fiber-reinforced composites using the 3D-version of the Puck failure criterion[END_REF].

Although it is preferred to consider a general 3D stress state for the damage evolution due to better performance as reported in [START_REF] Vasiukov | Non-linear material modeling of fiber-reinforced polymers based on coupled viscoelasticity-viscoplasticity with anisotropic continuous damage mechanics[END_REF][START_REF] Reinoso | A consistent anisotropic damage model for laminated fiber-reinforced composites using the 3D-version of the Puck failure criterion[END_REF], but the 3D Puck's theory demands additional iterative algorithms to search the plane of potential failure (Matthias Deuschle, 2012). Apart from that, the computational time increases significantly than the simpler phenomenological failure theory due to transformation of the stress tensor through increments of the angle range:[-90° to 90°] [START_REF] Thomson | Experimental and numerical study of strain-rate effects on the IFF fracture angle using a new efficient implementation of Puck's criterion[END_REF]. Consequently, here the plane stress format of Puck's theory (Puck, 2002b) is chosen and is discussed briefly.

There are total six basic failure mechanisms in laminated composites: matrix cracking in tension and compression, fiber/matrix de-bonding or known as shearing, Fiber

Fracture (FF) in tension and compression, and delamination (Falzon, 2011b). . In physical perspective, IFFs in Puck's theory reflect the meso-scopic matured transverse cracks along the whole thickness of a lamina. Such cracks are the outcome of the coalescence of the tiny matrix micro-cracking (cohesive fracture of the matrix) and fiber/matrix de-bonding (adhesive fracture of the fiber matrix interface). With the application of further loading, the IFFs get united and spread along the whole length or width of the single lamina aligned with the fiber direction in a composite coupon.

Eventually, each lamina attains the IFFs crack saturation level. Further application of the loading initiates other types of the damage such as diffused delamination in the adjacent plies [START_REF] Schuecker | A continuum damage model for fiber reinforced laminates based on ply failure mechanisms[END_REF], Schuecker, 2008a, Schuecker, 2008b). The mathematical formulation in eq.( 2.2) of Puck's theory (Puck, 2002b) for Fiber Fracture in Tension (FFT) and Fiber Fracture in Compression (FFC) have been taken from [START_REF] Reinoso | A consistent anisotropic damage model for laminated fiber-reinforced composites using the 3D-version of the Puck failure criterion[END_REF], and IFFs have been adopted from [START_REF] Lee | Initial and progressive failure analyses for composite laminates using Puck failure criterion and damage-coupled finite element method[END_REF] with the addition of replacing the stress components with the effective stress components computed through eq.( 2.1).

1 1 ; 0 1 1 ; 0 1 E FF T f T f E FF C f f C f T E IFF T vp T E f v v X E E f v v m X E Y f p S S Y σ σ σ σ σ σ σ σ σ +     = - - = ≥                 = - - = <                  = + -                   ( ) ( )
1 ; 0 1 ( ) 1 ; 0 1 ; 0 ( ) 2 1 vp E IFF C vp vp C E IFF C C vv p S f p p S Y f Y p S σ σ σ σ σ σ σ σ σ σ +  + = ≥   = + + = <           = + = <       - +                   (2.2)
f m σ is the stress magnification factor which imparts the effects of transverse stress on the fibers. Puck et al. (Puck, 2002b) have found that in case of plane stress, the results of the FFs based on maximum stress theory and the FFs computed by eq.( 2.2) differ only by few percent. However, if 22 σ is acting with considerable amount then it is recommended to use the FFs given in eq.(2.2). Similar signs of 11 σ and 22 σ imped the occurrence of FFs, and vice versa. The conditions for invoking each damage mode are listed in Table 2-1 against the failure exposure factor E f (Stress exposure factor is the ratio of the length of the stress state vector to the length of the strength vector in the Puck's failure envelope) with respect to each failure mode [START_REF] Lee | Initial and progressive failure analyses for composite laminates using Puck failure criterion and damage-coupled finite element method[END_REF].

, , E FF T f 11 0 σ ≥  FFT , , E FF C f 22 0 σ <  FFC , , E IFF T f 22 0 σ ≥  Mode-A , , E IFF C f 22 22 21 21
and 0

A vv C R σ σ σ < 0 ≤ ≤ τ    Mode-B , , E IFF C f 21 21 22 22
and 0

C A vv R σ σ σ τ < 0 ≤ ≤   

Mode-C

The slope points ( , , ) vp vp vv p p p + --in Puck's theory quantify the increase of the resistance to fracture due to the material friction in transverse compression, and vice versa in tension.

Puck et al. (Puck, 2002b) have defined range of these constants based on the experiments for CFRPs and GFRPs. The fracture resistance of the action plane denoted by A vv R can be computed from the in-plane strength parameters given by eq.( 2.3) [START_REF] Lee Cs | Initial and progressive failure analysis of composite laminates using Puck failure criterion and damage-coupled finite element methode[END_REF].

21 21 1 2 1 2 A C vv vp vp Y S R p S p - -   = + -       (2.
3)

The maximum value of the shear stress at which Mode-B switches to Mode-C is expressed by 21C τ . It is determined from the in-plane shear strength and the slope points of the Puck's envelope as given below [START_REF] Lee | Initial and progressive failure analyses for composite laminates using Puck failure criterion and damage-coupled finite element method[END_REF].

( )

21 21 1 2 C vv S p τ - = +
(2.4)

Damage accumulation functions

The thermodynamically consistent laws shall be used for the damage evolution which accepts the critical fracture energies in each direction for anisotropic materials as inputs.

Each damage variable is a monotonically increasing function and represents the irreversibility of various fracture mechanisms in composite laminate. The post-damage softening profile may be linear or exponential. The exponential damage evolution law is adopted from (Maimí et al., 2007a, Maimí et al., 2007b), and has been widely used [START_REF] Zhong | A continuum damage model for threedimensional woven composites and finite element implementation[END_REF][START_REF] Vasiukov | Non-linear material modeling of fiber-reinforced polymers based on coupled viscoelasticity-viscoplasticity with anisotropic continuous damage mechanics[END_REF][START_REF] Chen | Simulating progressive failure of composite laminates including in-ply and delamination damage effects[END_REF], Chen et al., 2012[START_REF] Reinoso | A consistent anisotropic damage model for laminated fiber-reinforced composites using the 3D-version of the Puck failure criterion[END_REF]. Accordingly, the damage state variables defined previously in eq.( 2.1) can be formulated as function of the Puck's theory evolving failure exposure factors with the applied load [START_REF] Reinoso | A consistent anisotropic damage model for laminated fiber-reinforced composites using the 3D-version of the Puck failure criterion[END_REF]. L is called characteristic element length, it depends on the mesh size. It can be retrieved from the FEM solver corresponding to the mesh size to the UMAT using CELENT which is one of the UMAT arguments in ABAQUS specifying characteristic element length c L . It is highly desirable to converge the FEM results for different mesh sizes up to a satisfactory degree. A mesh sensitivity analysis has been performed and is presented in the model validation section.

c j j j j E FF j j E FF j c j c j j j j E IFF j j E IFF j c j E IFF j E IFF j L X d A f A j T C f E L X L Y d A f A j T C f E L Y d A f f   = - - = =   -   = - - = =   -   = - -     2

Phenomenological plasticity modeling

It is deemed necessary to incorporate plasticity in the model in order to accurately predict the composite structure non-linear behavior particularly when the meso-scale based failure theory is used in CDM. The application of a very accurate failure theory will give reliable results until and unless it is used in collaboration with a suitable constitutive law which outputs the stress values to the former. Therefore, correct state of stress shall be computed which is only possible with the a more realistic constitutive law [START_REF] Vyas | Constitutive modelling of fibre-reinforced composites with unidirectional plies using a plasticity-based approach[END_REF]. Since the objective of the current work is to employ Puck's theory coupled with the plasticity, so a relatively simpler plasticity potential is adopted from [START_REF] Sun | A simple flow rule for characterizing non-linear Behavior of Fiber composites[END_REF] given by eq. (2.6), in order to be able to compare Puck's theory results

with the previously published results [START_REF] Chen | A combined elastoplastic damage model for progressive failure analysis of composite materials and structures[END_REF][START_REF] Weeks | Modeling the non-linear rate-dependent behavior in fiber-reinforced composites[END_REF] using the similar plasticity parameters involved in the coupled elastoplastic/damage model.

( ) ( ) ( ) 2 2 22 12 3 , 2 0 2 y f p a p σ σ σ = + - = σ      (2.6)
Here ( ) y p σ  is the yield limit which evolves with the load as function of the accumulated equivalent plastic strain denoted by p  : ( ) ( )

y p p α σ β = 
 , α , β are the hardening law exponent and coefficient, respectively. The parameter that establishes coupling between the transverse plasticity and in-plane shear plasticity is denoted by a. It shows the level of plastic strain developed under shear loading compared to the plastic strain developed in transverse stress. In other words, it describes the anisotropy in plasticity. The approach illustrated in [START_REF] Neto | Computational Methods for Plasticity: Theory and Applications[END_REF] will be followed for the plasticity modeling.

The quadratic form of the plastic potential can be expressed in effective stress space as rendered in eq.(2.7):

( ) ( ) 2 1 , . . 2 y F p p σ = - T σ σ σ      � (2.7)
Where is the mapping matrix and is given below. 0 0 0 0 3 0 0 0 6a

    =       � 2.2.5.1 Return Mapping Algorithm (RMA)
For the implementation of the model in an implicit scheme, the Return Mapping Algorithm (RMA) [START_REF] Simo | Consistent tangent oprators for rate-independent elastoplasticity[END_REF] is used. The model will be elucidated according to the steps necessary for RMA. At the beginning of an increment, the quantities (2.8) So using elastic constitutive law for trial stress computation we get:

. .

e e = + tr tr (n+1) (n+1) (n) σ ε σ Δε   = .
. in the plasticity function that implies:

( ) ( ) ( ) 2 ( ) ( ) 1 , . . 2 n y n F p p σ = - T tr tr tr (n+1) (n+1) (n+1) σ σ σ       (2.10) If ( ) ( ) , 0 n F p ≤ tr (n+1) σ  
then the current step is elastic and thus tr (n+1) σ 

is the same as the actual stress vector (n+1) σ 

. Following are the quantities to be updated for this elastic step in the UMAT:

1 n n p p + = = = tr (n+1) (n+1) tr (n+1) (n+1) p p (n+1) (n) σ σ ε ε ε ε      =  e .
Conversely, if ( )

( ) , n F p > 0 tr (n+1) σ  
then we have to solve eq.( 2.10) based on the returned/corrected stress (n+1) σ 

, and which is unknown for the current plastic step at the moment:

( ) ( ) ( ) 2 ( 1) ( 1) 1 , . . 2 n y n F p p σ + + = - T (n+1) (n+1) (n+1) σ σ σ       (2.11)
Now the prime objective is to express the actual/returned effective stress vector ( )

.

. . .

(2.12)

p (n+1) dε
is the differential in plastic strain vector and is given as

. = p p (n+1) (n+1) dε dε  
where  is the damage effect tensor defined previously in eq.( 2.1). The effective plastic strain increment p (n+1) dε  is determined by the normality rule:

.

F λ ∂ = ∆ ∂ p dε σ  
where λ ∆ is the nonnegative Lagrange's plastic multiplicator. It is important to note that when the plasticity function in eq.( 2.6) is used, then . Using the matrix notation, the increment in accumulated plastic strain dp  is shown as in eq.( 2.13):

( ) . . T dp = p p dε dε    � (2.13)
The mapping matrix  follows:

0 0 0 2 0 0 3 1 0 0 3a       =        
� Also, the normal vector F n to the scalar-valued plastic domain based on the associative flow rule assumption is defined as

. F ∂ = = ∂ F n σ σ  
 . The effective plastic strain vector is then read as

. . λ = ∆ p dε σ  
 . So eq.( 2.13) can then be transformed into:

( ) ( ) . . . . T dp λ = ∆ σ σ    �   (2.14)
eq.( 2.12) which relates the trial effective stress vector tr can also be shown:

. . .

e λ = -∆ tr (n+1) (n+1) (n+1) σ σ σ    � � . 
. And finally:

( )

1
.

.

e λ - = + ∆ tr (n+1) (n+1) σ σ    �  � . 
.  is the identity matrix. Putting the returned corrected stress vector (n+1) σ  in eq.( 2.11) we get eq.( 2.15):

( ) ( ) ( ) ( ) 1 1 2 ( 1) 1 = . . . . 2 T e e y n F p ∆λ ∆λ ∆λ σ - - +     + + -         tr tr (n+1) (n+1) σ σ     � �  � � .   .  (2.15)
Moreover, the scalar valued accumulated plastic strain for the current plastic step in the above expression can be computed as:

( ) ( ) ( 1) ( ) .
. . .

T n n p p ∆λ + = + (n+1) (n+1) σ σ       
(2.16)

The only unknown variable here in the scalar-valued function in eq.( 2.15) is the Lagrange's plastic multiplicator λ ∆ which can be solved using Newton-Raphson iterative scheme for finding the roots of the non-linear equation.

Newton-Raphson Iterative scheme

The th k guess for the solution of λ ∆ is obtained as:

( ) ( 1) ( ) k k k d λ λ λ - ∆ = ∆ + ∆ where ( ) k d λ ∆ is
given as follows:

( )

1 ( ) k k F d dF d λ λ λ - ∆ ∆ = - ∆
(2.17)

The derivative dF d∆λ in the above formula is determined next:

( ) ( ) ( ) ( 1) ( 1) 2 T y n y n d p dF p d d d σ σ λ λ λ + + = - ∆ ∆ ∆ (n+1) (n+1) dσ σ      (2.18) ( ) ( ) 1 1 . . . . . . e e e d λ λ λ - - = -+ ∆ + ∆ ∆ (n+1) tr (n+1) dσ σ    � � � �  � � .  .  .  (2.19) ( ) ( 1) ( 1) 1 ( 1) y n n n d p dp p d d α σ αβ λ λ + + - + = ∆ ∆    (2.20) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1)
+ + ∆ = + ∆ ∆   + ∆   ∆   ⇒ = ∆ (n +1) (n +1) (n +1) (n +1) (n +1) (n +1) (n +1) (n +1) (n +1) (n +1) (n +1) dσ σ σ σ σ σ dσ σ σ σ σ      � �   �    �    �           (2.21)
Put eq.( 2.19), eq.( 2.20) and eq.( 2.21) in eq.( 2.18) we get the complete derivative to be used in the Newton-Raphson formula in eq.( 2.17 ). The elastoplastic consistent tangent operator ep . is formularized in the upcoming section.

Consistent algorithm operator

In the implicit solver, the consistent algorithm operator is required to achieve quadratic . Following the procedure in [START_REF] Neto | Computational Methods for Plasticity: Theory and Applications[END_REF], the system of the equations for the RMA can be summarized as:

The elastic strain e (n+1) ε for the current increment can be shown in terms of the elastic trial strain e tr (n+1) ε :

. . 0

λ - + ∆ = e e tr (n+1) (n+1) (n+1) ε ε σ   � (2.22)
Similarly, the scalar valued equivalent plastic strain ( 1) n p +  and the plasticity surface in terms of the corrected stress vector for the current plastic increment can be written in the following way:

( ) ( ) ( 1) ( ) .
. . . 0

T n n p p λ + - -∆ = (n+1) (n+1) σ σ     �   (2.23) ( ) ( ) 2 ( 1) 1 . . 0 2 T y n p + -σ = (n+1) (n+1) σ σ     (2.24)
Differentiating eq.( 2.22), eq.( 2.23), eq.( 2.24) and writing the outputs in matrix form we obtain the following system of equation:

( ) 1 ( 1) 0 . 1 0 0 2 0 e n y dp d H λ λ ξ ξ λ σ - +   + ∆           ∆   - - =             ∆       -     F e tr (n+1) (n+1) T F T F n σ dε n n    �    .   (2.25)
Note that replace ( )

1 . e - = ε (n+1) (n+1) ε σ  .
while performing the above differentiation. Inverting eq.( 2.25) we get the desired operator to be used in the UMAT.

( )

1 1 2 1 2 ep e y λ λ σ ξ ξ - -   ∂ ∆ = = + ∆ + -   ∂ Η     (n+1) T T F F F F tr (n+1) Δσ n n n n Δε  �      . .   (2.26)
It is noteworthy that the above operator is non-symmetric; therefore, a non-symmetric solver shall be invoked during defining the step in ABAQUS. Here

. = F (n+1) n σ   , ξ = T F F n n  � , 1 ( 1) n p α αβ - + Η =   and (
)

1 e e - = =
 is the compliance matrix.

FEM implementation of the proposed model 2.3

The detailed formulation of the proposed model presented in the last section has been implemented in ABAQUS/Standard using the user defined subroutines UMAT with non-symmetric solver due to the fact that the elastoplastic consistent tangent operator ep . is non-symmetric. The sequence of the implementation has been outlined in the flow chart referred in Figure 2-1.

Results and discussion

2.4

For the validation of the model, two material systems of CFRPs with different ply orientation are selected for which the experimental data is available. Two study cases have been devised: (i) tensile tests on bi-angle balanced laminate coupons [START_REF] Weeks | Modeling the non-linear rate-dependent behavior in fiber-reinforced composites[END_REF], (ii) OHT specimens with different stacking sequences previously analyzed in [START_REF] Zhang | Progressive Damage and Failure Modeling in Fiber-Reinforced Laminated Composites Containing a Hole[END_REF][START_REF] Vasiukov | Non-linear material modeling of fiber-reinforced polymers based on coupled viscoelasticity-viscoplasticity with anisotropic continuous damage mechanics[END_REF], Lee et al., 2015, Chen et al., 2012, Maimí et al., 2007b).

Tensile tests on bi-angle balanced laminates

Damage models which use the energy release rates in the damage evolution are dependent on the mesh size during the post-damage loading phase. So, it is important to Calculate effective and nominal stress using elastic constitutive law:

• Update the Consistent Jacobian • update the State variables appraise the mesh sensitivity of the present model before going into the detailed FEM analysis of the full scale composite coupons.

Mesh sensitivity

Models that rely on the fracture energy for the damage evolution often possess the mesh sensitivity in a post-failure stress-strain. In other words, the finite element predictions do not converge to a unique solution as the mesh is refined because mesh refinement leads to narrower crack bands/crack width for a single finite element. This problem is frequently encountered in the localized cracking and failures. The characteristics element length c L is introduced into the damage models which vary according to the mesh size and hence mesh independent results is obtained up to a satisfactory degree (Maimí et al., 2007a, Maimí et al., 2007b). It is recommended to use elements of aspect ratio of 1 as discussed in the ABAQUS documentation (ABAQUS, 2013). The mesh independence of the energetic damage models has been demonstrated in (Maimí et al., 2007a, Maimí et al., 2007b[START_REF] Reinoso | A consistent anisotropic damage model for laminated fiber-reinforced composites using the 3D-version of the Puck failure criterion[END_REF], Falzon, 2011a, Falzon, 2011b[START_REF] Lapczyk | Progressive damage modeling in fiber-reinforced materials[END_REF].

Strain localization is a common problem in the damage accumulation causing material instability. The post-damage softening regime in stress strain curve depends on the size of the mesh if the crack band theory is not used proposed by Bazant et al. [START_REF] Bazant | Crack band theory for fracture of concrete[END_REF]. This theory ensures it by regularizing the energy dissipated f g computed during the solution by the characteristics length c L .

(energy per unit area) (energy per unit volume) ( )

f c g L mm =  (2.27)
Referring to eq.( 2.27), when the mesh is too coarse so c L increases and the value of f g becomes very smaller which is not physical. It is, therefore, of paramount importance that a proper maximum finite element size must be defined in the damage critical zone.

This maximum possible mesh size can be computed by comparing the strain energy per unit volume denoted by e to the fracture energy  per unit area Α of the fracture surface.

In order to initiate damage and the subsequent softening, the strain energy per unit volume must be equal or greater than the critical energy required per unit volume to fracture the material ( / ) c L 

. Interpreting this statement in mathematical form implies:

X E L E L X ≤   ⇒ ≤     ⇒ ≤ × ⇒ ≤ e σ ε    
(2.28)

The above equation will be used in all the simulations for determining the maximum suitable element size corresponding to the material constants. The characteristics length is also known as Crack band width. Referred to eq.( 2.5), the parameter j i A depends on c L and the materials constants. Keeping the materials constants fixed for a particular material, if the mesh size is varied from finer to coarser then eventually there will be a particular mesh size at which the exponential function in eq.( 2.5) remains no more stable and a numerical problem will occur in the simulations. Therefore, c L shall be calculated prior to launching the simulation for each material. The influence of c L on the damage growth in eq.( 2.5) is shown in Figure 2-2 where the damage initiates when the failure index becomes unity. It can be noticed that the coarser mesh, greater c L , tends to accumulate the damage faster and approaches towards the damage saturation rapidly that necessitates the critical consideration regarding the appropriate mesh size while damage analysis. The softening regime corresponding to these various meshes can be seen in Figure 23.

The crack band theory or the smeared crack approach, however, still possesses some extent of mesh dependency in the material softening due to the different damage growth as rendered in Figure 23.

Based on the above facts, therefore, a proper maximum possible element size shall be used to obtain relatively less mesh dependent results. The maximum possible finite element edge length max c L can be calculated using eq.( 2.29) for a particular anisotropic material (Maimí et al., 2007b). Cantilever Beam (DCB) [START_REF] Iwamoto | Intralaminar fracture mechanism in unidirectional CFRP composites[END_REF]. The shear fracture energy 21  can be assumed as the fracture energy in mode II which can be determined with 4-ENF test specimen [START_REF] Pinho | Physically based failure models and criteria for laminated fibre-reinforced composites with emphasis on fibre kinking[END_REF]. To check the mesh dependency of the present model, a cube of 1 mm was used in the same fashion as modeled in [START_REF] Reinoso | A consistent anisotropic damage model for laminated fiber-reinforced composites using the 3D-version of the Puck failure criterion[END_REF], Falzon, 2011b): individual ply thickness equivalent to the laminate thickness of 1 mm was provided in the conventional composite section in ABAQUS due to the fact that present model is based on plane stress assumption (conventional shell element: S4R) in contrary to the 3D models proposed in [START_REF] Reinoso | A consistent anisotropic damage model for laminated fiber-reinforced composites using the 3D-version of the Puck failure criterion[END_REF], Falzon, 2011b). The mesh sensitivity tests were performed for AS4/PEEK in fiber direction, transverse direction, and in pure shear case by varying the number of elements as depicted in Figure 2-3 (ac). The material properties of AS4/PEEK are presented in Table 2-2 [START_REF] Chen | A combined elastoplastic damage model for progressive failure analysis of composite materials and structures[END_REF], Soden et al., 1998b[START_REF] Weeks | Modeling the non-linear rate-dependent behavior in fiber-reinforced composites[END_REF]. Besides, the Puck's theory specific properties for CFRPs are listed in Table 2-3 (Puck et al., 2002) . It is evident from Figure 23) that the present model demonstrates minor mesh dependency during the stress softening regime in the material principal directions which is a desirable characteristic. A linear quadrilateral shell element S4R was used in all the simulations in the current work. Each element has an edge length of 0.5 mm with one integration point along the out-of-plane direction because damage is assumed uniform along the thickness direction in a single ply. The stacking sequences of the bi-angle laminates are selected with purpose that these possess a considerable degree of non-linearity and, thus, these demand the non-linear constitutive law. Despite some noticeable deviations from the target curve in case of [±30°] 2s , the model demonstrates the non-linearity with sufficient accuracy as compared to the experiments which is the result of the cumulative effect of both the diffused micro-damages and irreversible strain in the matrix. Hence equipped with this adequate elastoplastic constitutive law in collaboration with a more precise failure prediction based on Puck's theory, the breaking load of the OHT containing stress concentration is expected to be estimated with higher precision using the present model.

Numerical results of OHT specimens

The OHT ultimate breaking load is predicted for T300/1034-C according to the dimensions and symmetric boundary conditions shown in [START_REF] Zhang | Progressive Damage and Failure Modeling in Fiber-Reinforced Laminated Composites Containing a Hole[END_REF]. The material properties are presented in Table 2-4 which have been already reported in [START_REF] Chen | A combined elastoplastic damage model for progressive failure analysis of composite materials and structures[END_REF], Lee et al., 2015). The OHT breaking load prediction for the three laminates taken from [START_REF] Vasiukov | Non-linear material modeling of fiber-reinforced polymers based on coupled viscoelasticity-viscoplasticity with anisotropic continuous damage mechanics[END_REF], Chen et al., 2012) have been enlisted in Table 2-5 for the comparison with the experiments [START_REF] Sc | A Progressive Failure Model for Composite Laminates Containing Openings[END_REF]. Although the present model is based on the plane stress assumption, but still its accuracy is comparable to the 3D model developed by [START_REF] Vasiukov | Non-linear material modeling of fiber-reinforced polymers based on coupled viscoelasticity-viscoplasticity with anisotropic continuous damage mechanics[END_REF], and 2D model of Chen et al. [START_REF] Chen | A combined elastoplastic damage model for progressive failure analysis of composite materials and structures[END_REF].

This may be attributed to the accurate failure prediction of the Puck's theory for fiber as well as matrix in the FRPs as compared to Hashin's failure theory used in [START_REF] Vasiukov | Non-linear material modeling of fiber-reinforced polymers based on coupled viscoelasticity-viscoplasticity with anisotropic continuous damage mechanics[END_REF], Chen et al., 2012). It can be noticed that the relative error in the prediction of Chen model increases with the increase in the number of transverse plies [90°] in the laminates. This may perhaps indicates the inaccuracy of the matrix failure prediction based on the Hashin failure theory as was also emphasized in [START_REF] Lapczyk | Progressive damage modeling in fiber-reinforced materials[END_REF]. Similarly, The second configuration of the laminates where 24 plies are embedded has been chosen from [START_REF] Lee | Initial and progressive failure analyses for composite laminates using Puck failure criterion and damage-coupled finite element method[END_REF][START_REF] Zhang | Progressive Damage and Failure Modeling in Fiber-Reinforced Laminated Composites Containing a Hole[END_REF]. It is pertinent to describe that the model developed in [START_REF] Zhang | Progressive Damage and Failure Modeling in Fiber-Reinforced Laminated Composites Containing a Hole[END_REF] is micromechanics based, and hence it used the constituents' elastic properties as inputs rather than the homogenized UD ply elastic properties. As a result, the slope of the micro-model of Zhang et al. [START_REF] Zhang | Progressive Damage and Failure Modeling in Fiber-Reinforced Laminated Composites Containing a Hole[END_REF] The matrix cracking in the 45° plies initiated at the load magnitude of 14986 N. Then with further increase of the loading, the matrix meso-cracking propagated along the fibers, in the 90° and 45° plies, and ultimately attained its saturation level in the respective ply. The matrix damage accumulation in the 90° ply has been shown in Figure 2-8 which is similar to the model based on the 3D Puck's theory proposed in [START_REF] Reinoso | A consistent anisotropic damage model for laminated fiber-reinforced composites using the 3D-version of the Puck failure criterion[END_REF]. The declaration of the structure to be failed or safe depends on the application. If the leakage is the design criteria, for example in the petrochemical industry applications, then the onset of the IFF (Figure 2-8-a) may be the final failure.

However in other applications, these cracks are not considered as very harmful to the structural integrity. Still these not only degrade the stiffness but also provide access to the moisture which would be harmful for the structure in case of no adequate environmental surface protection. The ultimate failure of the laminate occurred due to the fiber breakage in the 0° plies at the load magnitude of 25392 N. This fiber damage evolution has been shown in Figure 2-9 at different load steps which followed the same pattern in the experiments and the model proposed in [START_REF] Reinoso | A consistent anisotropic damage model for laminated fiber-reinforced composites using the 3D-version of the Puck failure criterion[END_REF] that is, too, based on the same failure theory.

The fiber damage onset occurred at the proximity of the hole and then propagated perpendicular to the loading direction towards the free edges of the composite coupon. 2-7 with the experiments [START_REF] Chang | A Progressive Damage Model for Laminated Composites Containing Stress Concentrations[END_REF]. The simulation of the cross-ply laminate [(0/90) 6 ] s aborted prematurely like in the previous case which caused a higher relative error with the experimental data. The model predicted the bi-angle laminate [(±45) 6 ] s breaking stress with a very high accuracy (error of -1.4%) where the performance of the elastic/damage models is normally not satisfactory. This is can be noticed in Figure 2-6 where both the failure strength and the failure strain were underestimated significantly. The realistic alternative approach is to develop multi-axial failure theories that rely on the simplified uniaxial tests data.

The 3D available failure theories are elaborated in (Marklund, 2010), and in the Part B of WWFE [START_REF] Hinton | The background to Part B of the Second World-Wide Failure Exercise: Evaluation of theories for predicting failure in polymer composite laminates under threedimensional states of stress[END_REF] resistance with increase in transverse compression stress like the internal friction (Puck, 2002b). Consequently, the sustainable shear strength increases as long as the transverse compressive stress is increased. This peculiar capability makes Puck's theory ideal for the damage mechanisms study during the contact problems. The plane stress version was implemented previously in which the state of stress was computed based on the elastoplastic constitutive law. In the current chapter, the 3D formulation is discussed and finally will be implemented in collaboration with the damage evolution.

A detailed literature review about the Puck's theory and its implementation have been presented previously where most of the studies are limited to the plane stress case due to its simplicity. Furthermore, 2D Puck's theory is less expensive computationally because of the availability of analytical form which obviates the requirement of the fracture plane search algorithm. This searching of the fracture plane needs the stress transformation from the material local axis into the fracture plane axis in the angle range of [-90° to 90°] that results the simulation very costly particularly when larger number of finite elements are involved in the FEM domain. However, the plane stress assumption can lead to inaccurate damage initiation and evolution where the role of outof-plane stresses cannot be ignored. This inaccuracy is pronounced where there free edges, open holes, and notches exist in the composite structure [START_REF] Reinoso | A consistent anisotropic damage model for laminated fiber-reinforced composites using the 3D-version of the Puck failure criterion[END_REF]. Therefore, a complete 3D state of stress shall be computed and provided to the 3D Puck's theory in order to be able to predict a more realistic damage initiation, and the subsequent accumulation. More efficient algorithms for the search of the Puck's failure surface can also be used to accomplish more faster simulation, as for example, proposed in [START_REF] Schirmaier | A new efficient and reliable algorithm to determine the fracture angle for Puck's 3D matrix failure criterion for UD composites[END_REF]. Hence, it is highly advised to embed this searching algorithm for the sake of the time saving especially in case of the larger complex composite structures. 3D form of Puck's theory for the higher strain rate dynamic loading with efficient algorithm for fracture plane search has been implemented in [START_REF] Thomson | Experimental and numerical study of strain-rate effects on the IFF fracture angle using a new efficient implementation of Puck's criterion[END_REF].

Anisotropic damage constitutive law 3.2

Following the similar analogy as was adopted in the previous chapter, the effective stress vector σ  and Cauchy stress vector σ are mapped with the help of the damage effect tensor ( )

d 
which is in-tern a function of the damage variables as:

( ) . d = σ σ   (3.1)
More recently, 3D Puck's theory with the continuous damage evolution has been used in [START_REF] Reinoso | A consistent anisotropic damage model for laminated fiber-reinforced composites using the 3D-version of the Puck failure criterion[END_REF] with elastic constitutive law. The damage effect tensor is given as: axis of the single ply. To incorporate the unilateral loading effect in case of the cyclic loading these damage variables are described as

2 2 2 3 3 1 1 1 1 1 1 ( ) ; ; ; ; ; 1 1 1 1 1 1 f m m m m m d diag d d d d d d   =   - - - - - -     

Here

(1 )(1 )

T C f f f d d d = - - , 2 2 2
(1 )( 1) and3 3 3 (1 )( 1)

T C m m m d d d = - -
T C m m m d d d = - -
. Same sign conventions are used here as that of the original 3D

Puck's theory (Puck, 2002a) in the upcoming formulation for consistency with the 3D

Puck's theory available in the literature.

3D meso-damage onset functions

This section will cover the intra-laminar meso-damage initiation functions based on the 3D form of Puck's theory.

1 1 ; 0 1 1 ; 0 1 f f f E FF E F IF f F f E F E f v v E R E f v v m E R p f R R ψ σ ψ σ σ σ σ σ σ σ σ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ + - + ⊥       = - - + = ≥                 = - - + = <             = -            � � � � � � � � � � ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 t nt n1 n A A At 2 2 2 c t nt n1 , , n A A A A 1 ; 0 1 ; 0 n n E IFF n n p R R R p p f R R R R ψ ψ ψ ψ ψ ψ τ θ τ θ σ θ σ θ σ τ θ τ θ σ θ σ θ σ ⊥ ⊥⊥ ⊥ ⊥ - ⊥ ⊥ ⊥ ⊥⊥ ⊥ ⊥       + + + = ≥                          = + + + = <                           � � (3.2)
In [START_REF] Apruzzese | Numerical analysis of complex failure mechanisms in composite panels[END_REF], two discrete fracture angles were used: (i) when n 0 σ ≥  then fracture angle was assumed as 0° and the simplified form of the Puck's theory like in 2D from was used where the stress tensor in the fracture plane is identical to the stress tensor in the ply local coordinate system, (ii) similarly, for compression also a fixed angle of 53° was used and the stress tensor was transformed from local coordinate system into an angle of 53° and the stress exposure factor was calculated based on this transformed stress tensor acting on the fracture plane. Fixing these two discrete values obviates the searching and saves the computational time with significant amount.

However in the present model, the fracture angle will be searched as a continuous variable with a pre-defined step size. The fracture plane coordinates 1-t-n and the stress components on the fracture plane have been shown in Figure 3-2 .

Figure 3-2: Stress components in 1-t-n and 1-2-3 coordinate system (Puck et al., 2002) According to Puck's hypothesis, IFF is promoted by three stress components on the fracture plane. Whereψ is the angle between the shear stress components on the fracture plane and can be calculated as:

2 2 2 2 nt n1 2 2 2 2 nt n1 nt n1 cos ; sin τ τ ψ ψ τ τ τ τ = = + +       (3.
3)

The stress components on the fracture plane highlighted in Figure 3-2 can be computed by the transformation of the stress tensor from the ply local Cartesian coordinate system 1-2-3 to the fracture plane 1-t-n:

( ) ( ) ( ) 22
θ θ σ σ                 = - -                             (3.4)
It can be noticed in the above equation that all the stress components, except the stress in the fiber direction 11 σ , contribute their role in the matrix intra-laminar fracture which

t R R ⊥ ⊥ = , A R R ⊥ ⊥ = � � and ( ) c A c 2 1 R R p ⊥ ⊥⊥ ⊥⊥ = +
. Alternately, A R ⊥⊥ can also be computed by knowing the angle of pure transverse compression experimentally as:

( ) c A 2 1 tan R R θ ⊥ ⊥⊥ = +
. Here t R ⊥ , R ⊥ � and c R ⊥ are the tensile strength in transverse direction, in-plane shear strength, and compressive strength in transverse direction, respectively. Superscripts 'A','t' and 'c' signify the 'action plane', 'tension' and 'compression' respectively. When pure transverse tensile stress or pure in-plane shear stress is applied on a UD lamina then the fracture plane is coincident with that of the action plane, and hence the action plane strength is the same as that of the conventional strengths obtained by ASTM testing [START_REF] Matthias Deuschle | Finite element implementation of Puck's failure theory for fibre-reinforced composites under three-dimensional stress[END_REF]. In contrast, a single compressive stress cannot separate the material in its action plane rather the fracture plane is oblique (Figure 3-3). The failure is the sliding mode and hence is attributed to the resultant of shear stresses on the fracture plane: Moreover, the Puck' envelope inclination points are enlisted in Table 3-1 for CFRPs and GFRPs (Puck et al., 2002) obtained from extensive experiments. The generalized inclination parameters as a function of the angleψ in eq.( 3.2) are computed with the help of the following relations:

i i i 2 2 A A A cos sin ; , p i t p R c p R R ψ ψ ψ ψ ⊥ ⊥ ⊥⊥ ⊥ ⊥⊥ ⊥ + = = � � (3.5)
As far as the damage accumulation is concerned, similar damage evolution criterion is used here, as it is equally valid for the 3D state of stress that was used in chapter 2 for the plane stress model. It is appropriate to degrade the materials stiffness tensor in the oblique fracture plane axes recommended by [START_REF] Bogenfeld | A tensorial based progressive damage model for fiber reinforced polymers[END_REF] in cases when the fracture plane is at angle. But here the degradation in the stiffness tensor is accomplished by degrading in the material principal direction as suggested in [START_REF] Reinoso | A consistent anisotropic damage model for laminated fiber-reinforced composites using the 3D-version of the Puck failure criterion[END_REF].

Weakening effect of the Fiber Fracture (FFs) on on Inter Fiber Fracture (IFFs)

Experimentally, when a macroscopic FF is observed then it means that larger numbers of fibers have been fractured. Based on the statistical nature of the strength of fibers, some fibers having weaker strength may already have been fractured. Those local fiber fractures are regarded as local micro-damages which contribute in degradation: for instance, the fiber-matrix debonding is provoked by a broken fiber. This effect shall be considered at higher magnitude of 1 σ stress. The stress in fiber direction 1 σ does not interact with the stresses of the fracture plane but it reduces the strength against IFFs formation. As a consequence, the lamina is now more vulnerable to the formation of The IFFs calculated through eq.( 3.2) are enhanced as below due to the fiber direction weakening:

2 , , , 2
( ) ; 1 E IFF E FF E IFF w w f f s f a η η - = = - (3.6) 2 1 1 s a m - = - (3.7)

Implementation and validation 3.3

The model was implemented in ABAQUS via UMAT. The fracture angle in case of transverse compression was determined numerically. Then the unilateral effects and the cyclic loading in tension and compression were modeled. The motivation behind the 3D model was to use Puck's theory in the contact problem where it is best suited, thanks to the enhancement of sustainable shear strength as the transverse compression is raised in the second quadrant of the Puck's theory envelope. Therefore, the damage mechanisms induced by the indentation and adhesion wear of FRPs against the metallic ball is predicted in the final section of this chapter. In addition, the specific wear rate was also predicted in a qualitative manner for different angles of sliding with respect to fibers.

Fracture angle determination

A rudimentary study test case was simulated in ABAQUS to compare the fracture angle fp θ of the model with that of the analytical value for AS4-Epoxy chosen from [START_REF] Schirmaier | A new efficient and reliable algorithm to determine the fracture angle for Puck's 3D matrix failure criterion for UD composites[END_REF]. The potential plane is the plane which has maximum Puck's stress exposure factor corresponding to a given state of stress. This plane is determined by iterative process of varying the angle from -90° to 90° with a reasonable step size.

The exposure factors E f can be defined, in a general sense, as the as the ratio of the state of a pseudo stress vector { } σ  to the fracture vector (Matthias Deuschle, 2012).

{ } { }

( )

E fr f = σ σ   (3.8)
The graphical representation of this definition in the context of Puck's failure theory is shown in Figure 3-6 along with the angle variation.

Unilateral effects and stiffness recovery

It is a matter of fact that under uniaxial cyclic loading conditions the degradation mechanisms are quite complex to model. In such loading scenario, the opening and closing of previously formed micro-cracks as well as their interaction play a role in the stress-strain curve. Experimentally, it is observed that there is some recovery of the elastic stiffness as the load changes sign during a uniaxial cyclic test (Schuecker, 2008a). The stiffness recovery effect, also known as the "unilateral effect," is an important aspect of the material behavior undergoing through cyclic loading. This effect is usually more pronounced as the load changes sign from tension to compression, causing tensile cracks to close, which results in the recovery of the compressive stiffness.

The unilateral study was carried out for a single 8-node element C3D8 in the fiber direction and transverse direction. The load reversals from tension-compression-tension in the fiber direction are elaborated in Figure 3-9 denoted by the points 0-1-2-3-4-5-6-7-8. Once the fiber fracture in tension occurs (point 1) then the stiffness cannot be recovered in the subsequent compression and hence the same degraded stiffness shall be used (Maimí et al., 2007a, Maimí et al., 2007b). This is why that the slope of the line 0- Similarly, the load reversals were also modeled in the transverse direction.

Experimental observation illustrates that the crack in transverse matrix tension is brittle in UD lamina (Puck, 2002a). There are two approaches which are followed for the stiffness recovery in matrix cracks formation: (i) Cracks can be closed upon compression and the undamaged material stiffness is used. It interprets, the crack faces are in contact with each other and is capable to transmit load across the crack (Maimí et al., 2007a, Maimí et al., 2007b, Schuecker, 2008a[START_REF] Qing | 3D constitutive model of anisotropic damage for unidirectional ply based on physical failure mechanisms[END_REF], Falzon, 2011a), (ii) The degraded stiffness is used which is a more conservative approach [START_REF] Reinoso | A consistent anisotropic damage model for laminated fiber-reinforced composites using the 3D-version of the Puck failure criterion[END_REF] . Both cases are shown in Figure 3 Material is completely degraded at point 8.

Test cases of WWFE

The damage initiation for the bi-angle laminates are chosen for constructing the Puck's failure envelope: [±55°] 2s , [±65°] 2s , [±75°] 2s and [±85°] 2s under the uniaxial transverse tension and compression (Puck, 2002b). The material properties are given in Table 3-2 (Soden et al., 1998b). It can be noted from the results in Figure 3-11 that as the compressive transverse stress increases so the sustainable shear stress increases. It can be interpreted that the friction in the sliding surfaces is increased by the elevated transverse compression stress which is the actual case in experiments. This is a well-known interaction of 22 σ and 12 τ that must be considered. This kind of behavior cannot be observed while other classical criteria such as Hashin criterion [START_REF] Hashin | Failure criteria for unidirectional fiber composites[END_REF] is used. It was reported in [START_REF] Wang | Modeling of nonlinear response in loading-unloading tests for fibrous composites under tension and compression[END_REF] that most of the quadratic criteria give satisfactory results in the first quadrant where 22 σ is positive. The deficiency becomes pronounced when 22 σ is negative, therefore those criterion shall be used for failure analysis which consider the interaction. Furthermore, the failure envelope obtained from the experiments is not symmetric about the vertical axis of 12 τ due to this interaction. Based on these arguments, Puck's theory is ideal to be employed in the contact problem where 22 σ is negative in compression to exploit this interaction and predict more accurate damage exposure factors.

E E = (MPa) 0 0 21 13 G G = (MPa) v ⊥� t R � (MPa) c R � (MPa) t R ⊥ (MPa) c R ⊥ (MPa) R ⊥ � (MPa)

Application of 3D Puck's theory to contact 3.4 problem

Damage occurring in contact problem between FRP composites and metallic counterparts

Fiber Reinforced Polymers (FRPs), particularly thermoplastics (PEEK, PES, PEI, etc.) reinforced by carbon fibers, are used in the high temperature applications in tribocomponents where harsh operating conditions prevail such as aircrafts. Carbon fibers in the tribo-systems not only provide higher strength and thermal conductivity but also impart partial lubricity [START_REF] Bijwe | Carbon fabric reinforced polyetherimide composites: Optimization of fabric content for best combination of strength and adhesive wear performance[END_REF]. The damage induced by the contact of two counterparts and the subsequent wear in FRPs is of higher importance to study numerically. Nevertheless, this numerical analysis of wear process is a very complex task to perform. A diversity of fibers, both with respect to their properties and their fiber volume ratio, make the analysis of this process even more difficult. Although, it was suggested that the dominant wear process during the rubbing of fibrous composites is the adhesion [START_REF] Wieleba | The Mechanism of Tribological Wear of Thermoplastic Materials[END_REF], depending on the type of fibers and friction conditions, other wear mechanisms may also be present. However, in a set of fundamental factors that influence the wear of FRPs composite materials, the most important are fiber type, its fiber volume ratio in the composite as well as the structure of inter-phase between the fiber and the matrix. Here, we consider only the UD FRPs composites in which all the embedded laminae have the same fiber orientations in a laminate.

There is literature available that study the wear phenomenon in FRPs experimentally in abrasive wear and adhesive wear modes [START_REF] Lu | Wear and Friction of unidirectional carbon fiber-glass matrix composite against various counterparts[END_REF][START_REF] Bijwe | Carbon fabric reinforced polyetherimide composites: Optimization of fabric content for best combination of strength and adhesive wear performance[END_REF][START_REF] Cirino | The effect of fiber orientation on the abrasive wear behavior of polymer composite materials[END_REF]. [START_REF] Cirino | The effect of fiber orientation on the abrasive wear behavior of polymer composite materials[END_REF] investigated the dry abrasive-dominant wear in UD composite in which the composite was worn in three principal directions. These three directions were designated as N (normal), P (Parallel) and AP (Anti-Parallel) with respect to the fiber orientation as in Figure 3-12. The damage mechanisms were also studied caused by the abrasive or adhesive types of wear in each direction of sliding.

Here, we consider only the in-plane direction i.e. parallel and anti-parallel. To study the typical wear mechanisms, the scratched surfaces of the composite specimens were examined by Scanning Electron Microscopy (SEM) [START_REF] Friedrich | Polymer Composites: From Nano-to Macro-scale[END_REF]. Typical SEM micrographs can be seen in Figure 3-13 [START_REF] Friedrich | Polymer Composites: From Nano-to Macro-scale[END_REF]. In case of parallel fiber orientation, (Figure 3-13), shows the typical wear mechanisms that include: fiber/matrix de-bonding, fiber bending, fiber cracking, and matrix shear fracture. In Figure 3-13 right and left from the groove, the polished surface remained in its undamaged state. At higher magnification (see Figure 3-13(b)), a fiber crack transferring from one fiber to the next is noticeable. In addition, the formation of the shear fracture can also be found in the SEM photographs. to the compressive and frictional loads, the fibers bend in the depth and in the sliding directions, respectively. Both were associated with fiber fracture events in the center region of the core as well as in the transition between the groove and the undamaged area. Some of the broken fiber segments were removed out of the actual groove area, which can be a result of the additional torsion acting in the scratched material surface.

Repeated sliding of the counterpart may lead to cutting up of the fiber fragments at a later stage of the wear process. shows an enlarged segment of the scratched zone. Based on the figures, the typical wear mechanisms that can be observed comprise fiber/matrix de-bonding, shear fractures of the matrix material between the broken fibers (as already described for the P-fiber orientation), fiber cracking, and their removal from the fiber beds. By replacing the diamond tip indenter with a small steel ball, the characteristic wear and failure events are very similar but they do not appear so clearly.

Figure 3-14: (a) SEM-micrograph of a transversely oriented composite surface scratched by a diamond tip, (b) An enlarged segment of the contact area in the case of AP-fiber orientation [START_REF] Friedrich | Polymer Composites: From Nano-to Macro-scale[END_REF] Friedrich et al. [START_REF] Friedrich | Polymer Composites: From Nano-to Macro-scale[END_REF] described the presumed wear mechanisms of the UD FRPs composites in the form of wear cycles (Figure 3-15 and Figure 345678910111213141516). The wear damage follows the sequence as matrix wear, fiber sliding wear (also known as fiber thinning), fiber cracking, and wear by fiber/matrix de-bonding at the interface. The initial stage of the wear process includes matrix wear and fiber sliding wear. Following these two wear mechanisms, fiber cracking appears due to the matrix removal and the asymmetric contact pressure distribution produced by the sliding steel asperity. In addition, as a typical failure mechanism, fiber/matrix de-bonding occurs due to the shear and tension type loading. If fiber/matrix de-bonding has taken place, the local separation initiates additional fiber cracking, wear debris formation, and a more intensive wear process. In the steady state wear process, a so-called Compacted Wear Debris Layer (CWDL) covers the surface, composed of pulverized wear debris and matrix material. During the wear process this layer is continuously formed and removed by the surfaces sliding over each other. [START_REF] Lu | Wear and Friction of unidirectional carbon fiber-glass matrix composite against various counterparts[END_REF] in dryadhesive wear of glass matrix reinforced by carbon fibers with metals and SiC using pin-on-disc technique. In-plane bending was the dominant wear failure mechanism in the AP-direction where the longitudinal strains initiated the fiber fracture. In P direction, shear forces are dominant and causes fracture. The worn surface and the debris were also studied with SEM.

In the AP direction, the thickness of the debris layer was lower than the P-direction.

Most of the studies were limited to only three direction mentioned previously, but UD carbon reinforced thermoplastics (PEI) with 80% by volume were tested at range of angles, viz. 0°,30°,45°,60° and 90° in abrasive wear in [START_REF] Sharma | Influence of fiber orientation on abrasive wear of unidirectionally reinforced carbon fiber-polyetherimide composites[END_REF], and in dryadhesive wear in [START_REF] Sharma | Influence of orientation of long fibers in carbon fiberpolyetherimide composites on mechanical and tribological properties[END_REF]. The increase in the coefficient of friction mwas reported as the fiber angle with respect to the sliding direction increased. In addition, very low specific wear rate (volume removed per unit of work required) denoted by 3 0 ( / . ) K mm N m was determined for the sliding in the fiber direction. Four times higher specific wear rate was measured in the 90° sliding angle with fiber. The same results

have been obtained for epoxy matrix [START_REF] Ovaert | Wear of Unidirectional Polymer Matrix Composites with Fiber Orientation in the Plane of Contact[END_REF]. Slide orthogonal to fiber leads to higher wear and there are higher chances of the de-bonding, peel off the plies, and the pulverization of fibers. This was concluded based on the SEM observations in [START_REF] Sharma | Influence of orientation of long fibers in carbon fiberpolyetherimide composites on mechanical and tribological properties[END_REF] as given in Figure 3-17. Less work has been reported on modeling damage induced in the wear of FRPs. Most of the wear modeling is based on the micro-mechanics approaches considering multi-phase materials in a single ply [START_REF] Friedrich | Finite element analysis of a polymer composite subjected to a sliding steel asperity[END_REF][START_REF] Mzali | A micromechanical scratch model to investigate wear mechanisms in UD-GFRP composites[END_REF]. A sliding hemispherical asperity was modeled to correlate the damage modes encountered in the experiments of PEEK with 60% carbon fibers with the FEM results [START_REF] Friedrich | Finite element analysis of a polymer composite subjected to a sliding steel asperity[END_REF]. Slah et al. [START_REF] Mzali | A micromechanical scratch model to investigate wear mechanisms in UD-GFRP composites[END_REF] modeled the glass fiber and epoxy with Johnson-Cook (JC) behavior law [START_REF] Johnson | A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures[END_REF], and the fiber/matrix de-cohesion was assumed to follow the cohesive zone approach in ABAQUS/Explicit. It was urged in [START_REF] Mzali | A micromechanical scratch model to investigate wear mechanisms in UD-GFRP composites[END_REF] that the experimental approaches are insufficient for better comprehension of the wear behavior of composite materials. Hence, these shall be supplemented with the FEM.

Here, the wear and frictional contact problem in UD laminate is studied numerically based on the 3D Puck's theory in order to better understand the insights. The plies will be considered in one direction throughout the whole laminate, therefore delamination is not considered.

Finite Element Modeling

The present study aims the development of FE modeling that allows to provide explanations for different failure mechanisms based on the evaluation of contact stress and strain conditions produced by a sliding hemispherical indenter as illustrated in Figure 3-18 [START_REF] Friedrich | Finite element analysis of a polymer composite subjected to a sliding steel asperity[END_REF]. In particular, it is expected that this approach will give answers about the actual fiber stresses, matrix strains, events of fiber/matrix debonding, etc. The latter features will finally lead to the characteristic wear mechanisms of FRPs composites. (ASTM 2344) [START_REF] Sharma | Influence of orientation of long fibers in carbon fiberpolyetherimide composites on mechanical and tribological properties[END_REF]. The mechanical properties and strengths are listed in table Table 3-3. The particularity of this material is its bad adhesion between the fiber and the matrix as can be noticed by a lower value of the transverse tensile strength t R ⊥ .

Table 3-3: Engineering constants and mechanical strengths [START_REF] Sharma | Influence of fiber orientation on abrasive wear of unidirectionally reinforced carbon fiber-polyetherimide composites[END_REF] 11 The radius of the steel indenter which is modeled as a rigid hemisphere is Based on the convergence, a mesh with 223135 elements was defined. The mesh discretization was purposely refined close to the contact zone between the indenter and the composite and elements of edge size 0.005 mm were used.

E (GPa) 22 33 E E = (GPa) 21 13 G G = (GPa) 23 G (GPA) t R � (MPa) c R � (MPa) t R ⊥ (MPa) c R ⊥ (MPa) R ⊥ � (MPa)
0.25 R = mm.
The interactions between the indenter and the sample were controlled by the contact pairs' algorithm (surface to surface contact with slave surface type: node region).

Moreover, all the interactions were governed by the Coulomb friction model with an interfacial friction coefficient mof about 0.28 (for P-direction sliding ) and 0.3 (for APdirection sliding) based on the experimental studies of Sharma et al. [START_REF] Sharma | Influence of orientation of long fibers in carbon fiberpolyetherimide composites on mechanical and tribological properties[END_REF].

In this model, two steps were used to simulate the sliding of the indenter. Firstly, a vertical load of 1 N on the indenter up to the target penetration was applied against the sample. Secondly, the scratching began with the indenter tangential sliding until the target scratch length of 0.05 mm. The bottom face of the sample was constrained while symmetry conditions were applied to its both right and left sides.

Results and discussion

Contact results

A preliminary study in the first step was conducted to determine the contact parameters.

The aim was to verify the accuracy of the model in the indentation step before undertaking the complete wear problem. For this purpose, normal force was applied on the indenter. Following contact parameters were evaluated in this phase (see, Table 34• Total approach: indentation depth = δ

• Representative sizes of the contact area: Parallel to the sliding direction denoted by 2a and transverse to the sliding direction denoted by 2b

• Contact pressure profile and its maximum value: P max The contact parameters, shape and penetration, are in agreement with the results found in literature [START_REF] Friedrich | Finite element analysis of a polymer composite subjected to a sliding steel asperity[END_REF]. For AP-configuration, the penetration and the maximum contact pressure are slightly higher but it is trivial i.e. less than 10%. For isotropic materials, the contact surface is commonly circular but here it is elliptical due to the anisotropy in FRP composite plate. The contact results have confirmed the validation of the model and the BCs. Therefore, a detailed study of the stresses, damage mechanisms in wear and sliding are analyzed critically in the sequel.

Stress results

The Von Mises equivalent stress has been plotted for the two configurations in both the two steps of indentation and sliding as given in Figure 3-20. It can be comprehended that the Von Mises stress distribution for the two configurations are identical at 90° rotation in the indentation step. Though, due to the difference in the contact pressure during the indentation phase (see Table 3-4), a small difference of the maximum value in the stress is observed in between the P-direction and AP-direction.

In contrary, the Von Mises stress distributions are very different in the sliding steps for the two directions of sliding due to the additional application of tangential loading required to slide the indenter. Referred to Figure 3-20, the maximum Von Mises stress is located in the frontal part of the contact area in parallel (P) configuration, whereas in case of anti-parallel (AP) configuration, it is located at the rear side of the contact area.

It is necessary to analyze the stress components in the local material coordinate system 1-2-3 (1: fiber direction, 2: transverse direction, 3: direction normal to the contact surface) in order to get a well interpretation in regards of the damage and failure indices based on the failure theories which are considered here: Puck's failure theory, Hashin's failure theory.

In the following figures (Figure 3

Parallel configuration

Referred to Figure 3-21 and the stress contours of σ 11 and σ 22 there-in for Pconfiguration, it can be noticed that the fibers are subjected to bending. This bending is produced by the contact pressure and the corresponding transversal compressive stress σ 22 and axial compressive stress σ 11 under the contact zone. In the depth, transversal compression σ 22 decreases and eventually it transforms into traction. Similar stress patterns of σ 11 and σ 22 were reported during the indentation step in [START_REF] Friedrich | Finite element analysis of a polymer composite subjected to a sliding steel asperity[END_REF]. Higher tensile fiber transverse stresses are located at the vicinity of the contact zone. Due to the friction force, tension appears behind the contact area. As far as shear stresses are concerned, the most dominant shear stresses are σ 12 and σ 13 around 120 MPa for which the maximum value is located at the same position. For σ 12 shear stress, the maximum value is located on the surface at 45° direction behind the contact zone while the maximum of σ 13 shear stress is also behind the contact zone but located just below the surface.

Anti-parallel configuration

Under the contact zone (see Figure 3-21), for AP-configuration, the fibers are under compression in a thin subsurface which was found as less than 10 µm. Afterwards, the axial and transverse stresses, σ 11 and σ 22 respectively, change signs beyond this subsurface and become traction. Due to the friction force, compressive transverse stress σ 22 is located at the front of the contact zone while tensile transverse stress is obtained behind the contact zone. The shear stress σ 23 due to the friction is higher than the Pconfiguration having maximum value of 95 MPa as compared to the maximum value of 67 MPa for the sliding in the P-configuration. Similar to the P-configuration, the most dominant shear stresses are σ 12 and σ 13 having maximum value of 140 MPa and are located at the vicinity of the contact zone.

Based on the above stress analysis in the local material coordinate system, it is concluded that the stress magnitudes are higher in AP-configuration than the P configuration. This stress analysis is not sufficient to predict the nature of damage that take place when the rigid indenter is in contact with the composite specimen. Moreover, it is not very abstract to compare various stress component of P-configuration with APconfiguration until and unless a failure criterion is used that provide single scalar values by combining various stress component in a function. This function is known as stress exposure factor or damage exposure factor. For this purpose, an accurate failure theory is used to well understand the failure mechanisms that are induced during the contact of UD composite with indenter. Hence, the damage exposure factors are determined based on the Puck's criterion in the sequel.

Damage exposure factors analysis

The four stress exposure factors of the Puck's theory have been evaluated for P and APconfigurations. Remember that in the Puck's theory, IFFs exposure factors are linked with matrix damage and fiber/matrix de-bonding. On the other hand, FFs are interpreted as fiber damage. When a stress exposure factor turns into unity, so it implies that the particular damage is going to take place in the undamaged material. To highlight the influence of the sliding, the exposure factors plots, FFs and IFFs, during the indentation step and sliding step are given separately for parallel configuration in Figure 3-23 and Firstly, the damage exposure factors in indentation step are discussed for the two types of configurations. In indentation step, we remark that the same damage exposure patterns are found within a rotation of 90°. The difference in the values is due to the fact that the friction coefficient used for each configuration is not the same: 0.28 for the parallel configuration and 0.3 for the anti-parallel configuration. During the indentation step in both configurations, compressive damage of the fibers denoted by f E,FF,-and compressive/shear damage of the matrix expressed by f E,IFF,-appear in the contact zone due to the fact that the stress exposure factors exceeded one that implies that these types of damages have occurred. Furthermore, tensile/shear matrix damage f E,IFF,+ has also been induced in the indentation step, however this damage is located outside the contact area. Keeping in view the values of the exposure factors reached during the indentation step, fiber failure due to compressive stress is detected under the contact zone, fiber/matrix de-cohesion appears along the fibers in the local vicinity of the contact zone (f E,IFF,+ distribution), and compressive/shear matrix damage occur under and around the contact area with maximum values at its border (f E,IFF,-distribution). During the sliding step, the tensile fiber damage exposure factor (f E,FF,+ ) increased by a factor of 3 and also the distribution has changed as compared to the indentation step.

Parallel configuration

The maximum value of f E,FF,+ is higher than 1.04 (0.34 in indentation step) which is located behind the contact area, hence it indicates that fibers have failed in tension at the rear side of the contact area. Considering the fiber failure under compression (f E,FF,-), the exposure factor increased with 37.6% and the failure appeared at the frontal contact zone. As far as the matrix failure and fiber/matrix de-bonding is concerned, a similar pattern is obtained for the indentation and sliding steps with an increase of the exposure factor by 1.7 and 1.2 for f E,FF,+ and f E,FF,-respectively. The maximum values are still located at the border of the contact zone. Among all the exposure factors, a less increase was noted in f E,FF,-of approximately 17%.

A scratch test in the fiber direction produced clear features of fiber/matrix de-bonding as illustrated in Figure 3-25. In Figure 3-25 (a), the edge of the contact area is visible by the de-bonding lines. shows fiber fractures due to the high tensile stresses. In both figures, the formations of shear features of the matrix between the fibers are also noticeable. These findings are in good agreement with the FE results discussed in the present work. Anti-parallel configuration of the value of f E,FF,+ is lower than 1. As for the indentation step, compressive fiber failure occurs with a minor increase in the value of the exposure factor: 30%. However, f E,FF,-obtained a uniform distribution in the center of the contact area. The sliding force increased the fiber-matrix de-cohesion exposure index f E,IFF,+ by a factor of 6

approximately. We remark that the pattern is modified and the failure occurs behind the contact zone. 3-5. A significant increase in the damage exposure factors representing matrix cracks and fiber/matrix de-bonding in collaborative way in tension , f E,IFF,+ , is clearly seen as the angle increases. This can be easily explained by the fact that the considered material here has a low fiber-matrix interface strength ( t 7 R ⊥ = MPa). A non-monotonic evolution is observed in the matrix damage caused by compression and shear (f E,IFF,-): an initial increase following by a decrease for a maximum value reached at an angle of 60°. On the other hand, an inverse tendency is observed for tensile fiber damage exposure factor (f E,FF,+ ). A slight decrease is also obtained for the compressive fiber damage exposure factor (f E,FF,-). [START_REF] Sharma | Influence of orientation of long fibers in carbon fiberpolyetherimide composites on mechanical and tribological properties[END_REF] conducted experiments using pin-on-disc method for different offaxis angles of the fibers. The damages exposure factors are determined for the experiments of Sharma et al. [START_REF] Sharma | Influence of orientation of long fibers in carbon fiberpolyetherimide composites on mechanical and tribological properties[END_REF] in the upcoming section.

defined the specific Puck damage exposure factor as the ratio between the damage exposure factor and the applied normal load.

Figure 3-29: Specific wear rate as a function of the fiber orientation angle [START_REF] Sharma | Influence of orientation of long fibers in carbon fiberpolyetherimide composites on mechanical and tribological properties[END_REF] In the case of a plane square pin, the highest exposure factor is obtained for the matrix damage under compression and shear mode which is expressed as f E,IFF,-. In decrease with the applied load. In other words, severe the conditions of loading, better is the tribo-performance of composites. Hence, more the normal applied load more is the wear resistance. Panier et al. [START_REF] Panier | Finite Element Analysis of the damage occurring in the sliding contact between a metallic ball and a composite plate[END_REF] showed that with Hashin's criterion this tendency cannot be explained in FEM modeling as shown in Figure 3-32.

Consequently, Puck's criterion is employed here which is able to predict the experimental trends of the increase in wear resistance by increasing the normal load. In fact, Puck's criterion takes into account that the materials shear strength increases with the increase in the compressive transverse stress σ 22 . This implies that due to the enhanced friction in the materials surfaces during higher transverse compression, the material is able transfer more load as compared to the lower transverse compression. So more the contact pressure is higher more the shear strength grows. Of course, there exists a critical limit of compressive σ 22 beyond which no increase in the shear strength is possible (Puck et al., 2002, Puck, 2002b). The shear strength then starts decreasing after crossing the critical limit by the compressive σ 22 . This can be better The trends showed better correspondence as the orientation of the angle between the fiber and the direction of sliding is varied. In regards of tribological performance, the orthogonal sliding with respect to fiber resulted into higher damage exposure factors and hence higher wear rate. This damage model can be used in impact modeling and bending problems. The model needs further enhancement to use the non-linear constitutive law along with delamination modeling based on the traction-separation law.

Introduction 4.1

This chapter discusses the experimental characterization of the composite material system used, and their fabrication in the current study. In order to predict the structural response of the composite materials in the service conditions, the engineering constants, and the strengths shall be known to the designer. Unlike metals, due to anisotropy the properties shall be determined in various directions, and shall be determined carefully as per the designated standards. Strength and failure characteristics of FRPs under combined effect of in-plane shear and tensile normal stresses are of great importance for effective design. The reason for this interest is that composite laminates, in general, are weak in shear loading and when combined with tensile stress, the shear strength deteriorates even further. For this purpose, Arcan testing of the bi-angle laminate will be presented at the end after the preliminary UD characterization in the material principal directions.

Materials specifications and specimens 4.2 preparation

The laminates studied here are composed of the carbon fibers and epoxy which is a thermosetting resin. The specifications of each constituent have been summarized in Table 4-1. The carbon fiber based UD dry fabric of width 1.8 m was procured which has in-plane polyester stitching only for the sake of the consolidation of the dry fabrics, and the ease in handling. The carbon fiber reinforcement was 154 g/m 2 whereas the polyester yarn was 4 g/m 2 . The dry fabric and the stitching have been shown in Figure 4-1.

The required composite plates were fabricated with the resin-infusion method in the desired direction of the fiber orientation using the resin specialized for the infusion having low viscosity: Epoxy (SR 8100), Hardener (SD 8824) with mixing ratio 100/33. Proper care was taken to prevent the air bubbles formation and trapping in the resin during infusion and the subsequent polymerization.

Mechanical characterization 4.3

The mechanical characterization of the composites is highly important from the design and analysis as well as the life prediction point of view. Due to anisotropy and heterogeneity, the characterization of composites is somehow difficult unlike the metals. The key considerations regarding the mechanical testing include:

• Specimen must exhibit the desired state of stress and strain and the distribution must be uniform in the gauge section.

• Minimization of the cost of material, fabrication and testing.

• Specimens used to generate the data must be of same fiber volume fraction and must have been fabricated with the same fabrication process.

• The geometries of the testing composite coupons must be simpler to realize into their shapes. Usually, flat rectangular topologies are preferred.

The required composite coupons sizes were cut according to the applicable ASTM standard for each type of the characterization test. Flat rectangular specimens were cut with precaution to avoid any damage for all the tests with the help of a cutting wheel machine, SSCM 800 MATRIX, equipped with water as coolant to prevent heating and the associated thermal deformation and residual stresses. Coolant is also necessary to prevent the carbon contamination in the air which is hazardous to human health. The LLOYD LR50K Plus tensile machine with load cell of 50 kN was used in mechanical testing with the crosshead speed set at 1 mm/min for the tests in fiber direction and 2 mm/min for the tests in off-axis direction. All the tests were conducted at displacement controlled mode at room temperature.

Longitudinal and transverse tensile tests

The uniaxial tensile test is the most widespread and most studied test because it is easier and it estimates the major load carrying capability of the laminate along the fibers and transverse to the fibers. The well-known objectives of these tests are to measure the ultimate tensile strength, transverse tensile strengths and the respective elastic moduli in each direction. However, the major Poisson ratio can also be measured with additional instrumentation. The relevant standard for the tension tests in both orientations of the FRPs is ASTM D 3039/D for the 0° and 90°, the dimensions complying this standard are illustrated in Figure 4-4. Total ten specimens were tested for the 0° orientation of the fiber with the extensometer having gauge length of 25 mm was installed on five out of ten specimens. The laminates were embedded with six plies in order to achieve 1 mm thickness. The extensometer was removed at the later stages of the loading with the purpose to protect it from the damage. For this on-axis loading the value of 1 mm/min was set in the displacement controlled fashion. The test set-up with the extensometer installed on the specimens and some of the post-test fracture specimens are presented in Figure 4-5. Linear elastic behavior and abrupt catastrophic failure was observed without any yielding/softening, as expected for 0°, where most of the specimens failed near the tabs. The material behavior in transverse tension direction is also linear as in Figure 4-8 which has also been obtained in numerous other studies as for instance in (Van Paepegem et al., 2006a). The materials properties in the transverse direction are enlisted in Table 4-3 in which failure strain has been extrapolated after the removal of the extensometer from the specimen in each test.

In-plane shear test

There are several methods for the in-plane shear testing of composite materials.

Examples are: (i) At an angle of ±45° symmetrically laminated about the midplane, the transverse strain 22 ε is very small and the shear strain 12 γ is about 13 times than the fiber direction strain 11 ε [START_REF] Olsson | A survey of test methods for multiaxial and out-of-plane strength of composite laminates[END_REF], Van Paepegem et al., 2006a). (ii) an off-axis angle of about 10°, the transverse normal strain 22 ε is almost zero and the shear strain 12 γ is about 3-4 times the strain in the fiber direction 11 ε . This testing is accompanied with some difficulties like the loading clamps and requirement of longer specimen causing bending. Paepegem et al. (Van Paepegem et al., 2006a) obtained considerable lower strength using this method than the ±45° laminate.

Apart from the above set-up, other two experimental methods for the in-plane shear characterization include a V-notch rail shear test (ASTM D7078-12), and V-notch beam test method also known as Iosipescu shear test (ASTM D5379-12) as followed in (Tan, 2016) for CFRPS, and in [START_REF] Khashaba | In-plane shear properties of cross-ply composite laminates with different offaxis angles[END_REF] for in-plane loading at angles ranging from 0° to 90°, but these require specialized fixtures. The inter-laminar (out-of-plane) shear strength is determined also with two tests: (i) Short beams test (ASTM D2344-13), (ii) double V-notch test. All these four characterizations tests have been discussed in detail in [START_REF] Almeida | Effect of fiber orientation on the shear behavior of glass fiber/epoxy composites[END_REF]. The in-plane tensile testing of a symmetric ±45° laminate for FRPs is widely adopted in accordance with ASTM D3518/D3518M (the equivalent in ISO 14129:1997) because it is an easier method to characterize the material in shear. In this test, the laminate is loaded along the global 0° direction and the stress is taken as half of the longitudinal stress recorded in the experiment. It shall be noted that the state of stress is not the pure shear. The normal stresses in the fiber direction and transverse to the fiber direction also exists and thus affect the results by interacting [START_REF] Arcan | A Method to Produce Uniform Plane-stress States with Applications to Fiber-reinforced Materials[END_REF]. Nevertheless, this tension test of ±45° laminate is frequently followed in the shear characterization of FRPs due to its easiness.

±45° in-plane shear test (ASTM D3518/D3518M)

The specimen is shown in Figure 4-9 where the length is 175 mm and width is 25 mm.

Total 8 plies were embedded in laminate of [±45°] 2s to maintain a thickness of 1 mm.

Tensile load was applied at displacement rate of 2 mm/min. Digital Image Correlation (DIC) was used to record the strain field. Alternately, strain gauges can also be installed to measure the strains in the on-axis and transverse direction. Independent strain gauges shall be installed in the directions of 0°, 45° and 90° instead of rosettes which is not appropriate due to considerable difference between the three strains which result into loosening and distortion of the rosettes (Van Paepegem et al., 2006a). Where the load at the i th point on the load-displacement curve is denoted by i P . In eq. tests, the fibers try to align with the applied load known as scissoring effect due to which the final failure takes place at higher loads. This is the reason why ASTM has defined the shear strength to be taken at 5% strain no matter the ultimate failure occurs at the higher stress or strain.

The in-plane shear stress-strain curve is given in Figure 4-11 where a higher degree of non-linearity can be noticed. The end point in this shear stress-strain curve does not correspond to the failure point of the specimen because all specimens ruptured well beyond the strain shown in this figure. The ultimate shear stress measured was 77.71

MPa at complete rupture. According to ASTM D3518-21 failure strength in shear shall be selected at 5% cut-off strain shown by the dotted line. For the 2D model developed in chapter 2, the plasticity parameters were also extracted by inverse FEM method for the materials fabricated here. Table 4-4 enlists the shear properties and the fitting parameters pertaining to the plasticity section of the model. Total four specimens were tested and one specimen marked as number 4 is presented in Figure 4-12 after the test along with the strain contours acquired through DIC system.

The maximum fracture strain in y-axis is 15 % which can also be read in the longitudinal stress presented previously in Figure 4-10. MPa. This stress is computed using the net section of the OHT specimen where diameter of the hole is 6.35 mm. Non-linearity in both the tests was observed. In case of [0/90°] 4s laminate, this nonlinearity is minor and may be regarded as due to the progressive matrix failure in 90°.

On the other hand, the cause of non-linearity in [±45°] 4s is due to both the irreversible strains in the matrix material (progressive diffused damages and meso-scale damages)

as well as plasticity in the matrix material. The ultimate failure in [0/90°] 4s was observed due to the fiber breakage whereas mixed type of failure was witnessed describing both matrix and fiber failures in [±45°] 4s OHT.

In-plane combined tension/shear loading in 4.4 composites

Olsson [START_REF] Olsson | A survey of test methods for multiaxial and out-of-plane strength of composite laminates[END_REF] discussed the available testing methods for composites which circumscribe the in-plane and out-of-plane loading with pros and cons such as tubular specimens, cruciform specimens, Arcan specimens, flexure testing, etc. The main advantage of tubular specimens is the absence of the free edge effect and it covers most of the failure envelope quadrants. But the major demerits are the fabrication of the tube, controlling the manufacturing quality, buckling, the experiments may be difficult to manage keeping in view the hydraulic connections, leakage, etc., to exert pressure [START_REF] Smits | Design of a Cylindrical Specimen for Biaxial Testing of Composite Materials[END_REF]. Cruciform specimens can be employed for the in-plane bi-axial normal stresses and are comparatively easier to fabricate. The complicated factors are the extra machining to get a reduced thickness at the center and fillets to restrict the failure within the gauge section. In addition, testing rig needs four orthogonal loading actuators. The Arcan specimen can only be used for the in-plane shear with tension or compression whereas in case of compression a fairly complex supporting rig is required as shown in Figure 4-18 [START_REF] Olsson | A survey of test methods for multiaxial and out-of-plane strength of composite laminates[END_REF]. The shear testing with Arcan of AS4/PEEK in (El-Hajjar, 2004) indicated a higher strengths than the conventional off-axis test of ±45°.

Therefore, shear characterization of FRPs that employed Arcan fixture is considered also an important method. Besides from the in-plane application of the loading on various types of composites, Arcan fixture has been used in many studies for out-of-plane quasi-static testing and delamination as well [START_REF] Alfonso | Determination of the 3D failure envelope of a composite based on a modified Arcan test device[END_REF][START_REF] Cognard | A modified Arcan test to analyze the behavior of composites and their assemblies under out-of-plane loadings[END_REF][START_REF] Gning | Through-thickness strength measurements using Arcan's method[END_REF].

Moreover, the Arcan fixture has bright prospects for exerting the bi-axial fatigue loading of normal stress in combination with shear stress as reported in [START_REF] Li | Shear Fatigue Test and Life Prediction of Composite Laminates Trans Tech Publications[END_REF] that has been less studied as compared to the bi-axial normal fatigue studies using the cruciform specimen investigated in [START_REF] Sawadogo | Experimental and Numerical Study of the Fatigue of GFRP Composites under Complex Loadings[END_REF][START_REF] Smits | Design of a cruciform specimen for biaxial testing of fibre reinforced composite laminates[END_REF]. Arcan test fixture was modified here in this study and was used for in-plane shear combined with tension loading.

In-plane bi-axial loading with Arcan 4.5

The Arcan fixture was initially conceptualized by Arcan and his co-workers for the shear testing of polymers and composites by utilizing a specially S-shaped composite specimen [START_REF] Arcan | A Method to Produce Uniform Plane-stress States with Applications to Fiber-reinforced Materials[END_REF]. The S-shaped specimen was later transformed into the butterfly shaped specimen by [START_REF] Arcan | A method to produce uniform plane-stress states with applications to fiber-reinforced materials[END_REF]. The main motivation behind the work of Arcan et al. was to make available an alternative testing facility in place of the pure shear testing through the torsional loading of the circular tube specimens due its cost, difficulty in fabrication, and prone to error in the fiber alignment in addition to the gripping problem [START_REF] El-Hajjar | In-plane shear testing of thick-section pultruded FRP composites using a modified Arcan fixture[END_REF]. It was demonstrated based on the photoelasticity and the IR thermography that pure shear stress state can be achieved in the significant section of the Arcan specimen that is good enough to glue strain gauges.

The conventional methods of in-plane shear characterization that include V-notch railshear test, Iosipescu shear test, and the tensile test of ±45° possess some disadvantages like the normal stresses are not zero and thus the results are less reliable in regards of denying the pure shear elastic constants. This alternative method was critically scrutinized in [START_REF] Hung | An Evaluation of the Arcan Specimen for Determining the Shear Moduli of Fiber-reinforced Composites[END_REF] for the determination of the longitudinal and transverse shear behavior of AS4/PEEK. It was highlighted that due to the compact and shorter geometry of the specimen, no bending problem was observed and at the same time it can be used to get shear properties in all orientations. The twisting and bending checks were accomplished by mounting the gage rosettes on both sides and negligible bending effects were observed. Thus specimens can be cut in all directions to better characterize the shear anisotropy: the shear moduli 12 G , 21 G , 31 G , 32 G were determined experimentally and the longitudinal moduli ( 12 G , 21 G , 31 G ) were found with 1% error with each other.

Consequently, the shear properties are more reliable due to the absence of the normal stresses interaction and bending, in contrary, to the other counterpart in-plane shear testing methods. The modified Arcan was employed in [START_REF] De Sousa | Failure prediction of hybrid composite using Arcan's device and Drucker-Prager model[END_REF] for the anisotropy quantitative study of the randomly oriented hybrid composites at various angles. Tan et al. have investigated and characterize the damage caused by the in-plane combined loading of shear with either tension or compression in rectangular specimen having sharp notch and blunt notch in FRPs [START_REF] Tan | Failure mechanisms of a notched CFRP laminate under multi-axial loading[END_REF][START_REF] Tan | The effect of laminate lay-up on the multiaxial notched strength of CFRP panels: Simulation versus experiment[END_REF], Tan et al., 2016).

Modifications proposed in the Arcan fixture

The original Arcan fixture was modified by Yen et al. [START_REF] Yen | Evaluation of a modified Arcan fixture for the in-plane shear test of materials[END_REF], and later used also in [START_REF] El-Hajjar | In-plane shear testing of thick-section pultruded FRP composites using a modified Arcan fixture[END_REF] for the shear characterization and stress-strain behavior. The specimen is sandwiched in four steel parts and its orientation is fixed relative to these halves. The whole system is integrated with the loading machine with help of clevis pins to reduce out-of-plane forces and moments. In the present study, the Arcan fixture used in the previous work [START_REF] Yen | Evaluation of a modified Arcan fixture for the in-plane shear test of materials[END_REF][START_REF] El-Hajjar | In-plane shear testing of thick-section pultruded FRP composites using a modified Arcan fixture[END_REF][START_REF] Tan | Failure mechanisms of a notched CFRP laminate under multi-axial loading[END_REF][START_REF] Tan | The effect of laminate lay-up on the multiaxial notched strength of CFRP panels: Simulation versus experiment[END_REF], Tan et al., 2016) has been modified further to improve the performance. In contrary to the previous fixture, the attachment points with the loading machine have been fixed, hence, there are only two fixed pin attachments. Instead, the specimen can now be rotated at the desired angle without detaching the fixture repeatedly from the tensile machine in case of angle change. The newly designed fixture is appropriate for fatigue tests also because the classical design of Arcan fixture is not stable enough to realize the cyclic tests at higher frequency.

The CAD model of the modified Arcan fixture is shown in Figure 4-19 with different loading angle in order to apply multi-axial loading and, the specimen is in the middle so that the applied load is at the mid-plane of the specimen. The dotted line AB shows the notch section and when load is applied parallel to the notch section then a state of pure shear is achieved. Similarly, when load is exerted at 90° to the notch section then normal tensile state of stress is simulated experimentally. The combined shear and normal tension state of stress is established when the angle between the notch section and applied load varies in between 0° and 90°. All these loading condition are illustrated schematically in Figure 4-19.

Aluminum Alloy AA6061 was used for the two squared identical halves rather than the four round steel parts to minimize the machining cost and efforts. The range of angle variation has been improved as compared to the previous fixture: the specimen can be rotated at an angle up to 90° with 15° step which enhanced the capability to exert multiaxial loading with different combination of shear/tension ratios. Arcan specimen is mounted on the fixture with total six M5 high grade bolts far away from the gauge section.

Combined bi-axial loading

Arcan testing was first time performed in case of bi-axial loading by Voloshin and Arcan [START_REF] Voloshin | Failure of Unidirectional Fiber reinforced Materials--New Methodology and Results[END_REF] after its initial applications for the characterization of the pure shear properties. Adhesives were also qualified through testing in combined loading of shear based on the Arcan configuration [START_REF] Liechti | On the Uniformity of Stresses in Some Adhesive Deformation Specimens[END_REF]. The versatility of the Arcan fixture is that it can be used for in-plane uniaxial normal loading, shear loading separately and, or combined loading of shear with either tension or compression [START_REF] Hung | An Evaluation of the Arcan Specimen for Determining the Shear Moduli of Fiber-reinforced Composites[END_REF] very easily by rotating the specimen at the desired angle of interest. This makes Arcan a very robust and flexible technique in regards of accommodating the fiber offaxis tests. On the other hand, if conventional testing is used then the specimens for each angle would be required to fabricate while in Arcan a UD laminate can be aligned readily. While loading an off-axis ply is always accompanied by rotation generated by the shear component of the loading, there is always a problem and it needs specialized clamping devices to permit rotation. Therefore, Arcan offers a more user friendly testing facility with less scatter in readings which can cause by fabricating the off-axis plies and error in the fiber alignment.

Parametric optimization of the Arcan specimen

The advantage of Arcan fixture is the fairly uniform stress distribution in the test section but it is dependent on the fiber direction with the applied load axis [START_REF] Olsson | A survey of test methods for multiaxial and out-of-plane strength of composite laminates[END_REF]. So, uniform shear distribution in Arcan fixture can only be ascertained if an optimized sample is used for each stacking orientation through the iterative FEM procedure. The size of the optimized Arcan sample depends on the stacking orientation of the intended composite to pass through the characterization plan. The optimization study was carried out in [START_REF] Hung | Finite Element Analysis of the Arcan Specimen for Fiber Reinforced Composites under Pure Shear and Biaxial Loading[END_REF] on the basis of the linear elastic FEM to consider the fiber orientation, notch root angle and notch radius. In addition, [START_REF] El-Hajjar | In-plane shear testing of thick-section pultruded FRP composites using a modified Arcan fixture[END_REF]) also investigated the parametric analysis but using the modified version of Arcan proposed by Yen et al. [START_REF] Yen | Evaluation of a modified Arcan fixture for the in-plane shear test of materials[END_REF] The objective of the parametric study is to get a uniform shear at the central portion of the significant portion (gauge length) equal to or greater than the dimensions of the strain gauges, so that gauge rosettes can be successfully installed. Area between the notches in Arcan specimen is called significant section. When a load is applied at an angle to the significant section then a combined normal and shear stress condition is produced. The Arcan specimen, the uniform shear stress distribution and the corresponding shear stress-strain plots produced in (El-Hajjar, 2004) are given in Figure 4-23. The direct stress components are also nearly equal to zero as plotted in Figure 4-26-a.

Following this figure, more than 60% of the cross section has a constant shear stress distribution. This optimized specimen will be used in the bi-axial testing having notch radius of 2.38 mm and angle 102.7°. The complete experimental set-up which includes the data acquisition system from two perpendicular strain gauges installed on each specimen is given in Figure 4-29.

Arcan experiments 4.6

Experimentally, bi-axial loading can be considered at two different levels: (i) External level, (ii) internal level. The external level is to apply the external loading in two axes which will induce a bi-axial stress inside the material: examples, (i) a cruciform specimen [START_REF] Sawadogo | Experimental and Numerical Study of the Fatigue of GFRP Composites under Complex Loadings[END_REF][START_REF] Smits | Design of a cruciform specimen for biaxial testing of fibre reinforced composite laminates[END_REF] which is only limited to apply biaxial direct stresses and incapable to apply shear. Though, shear stress can be obtained by changing the fiber rotation in cruciform tests as well. (ii) using a tubular specimen which has the capacity to apply internal pressure, external pressure, axial load and torque in combination to cover the entire range of the failure envelope. The second method is to exert the load along one axis but at an angle which resolves the applied load into shear and direct stress components, thus, a bi-axial state of stress is induced. The Arcan testing system is corresponding to the latter case i.e to induce the bi-axial stress at internal level. The variation of internal stress components 11 σ , 22 σ and 12 τ with fiber angle is illustrated in The shear strain field recorded with DIC is shown in Figure 4-34 (a) at a certain load step in order to highlight the uniformity in shear strain.

Numerical Simulations

The pure shear case of cross-ply experiments have been simulated using the 2D elastoplastic model proposed in chapter 2. The model uses the input parameters given in It shall be noted that the elastic constants and the plasticity parameters are determined experimentally here while the damage critical fracture energies required for the damage evolution in the material principle directions were chosen from chapter 2 for AS4/PEEK. The strengths in compression were also assumed. The right side three holes were fixed by defining the fixed displacement boundary conditions (BCs) on the three reference points which were linked with the circumferential nodes on the hole by specifying beam constraints. For the shear load application, three reference points were also defined with beam constraints as was done for the displacement BCs. Displacement load in y-axis were applied on the three reference points while the other two components of the displacements were kept zero to avoid bending. The boundary conditions and load points are explained in Figure 4-37 in addition to the meshing. A 4-node plane stress with reduced integration formulation SR4 was used. The shear strain distribution at failure is compared with DIC in Figure 4-39. The model under estimated the failure shear strain. It was noted in the FEM simulations of crossply Arcan specimen that the shear strain distribution was similar in magnitude in the 0° as compared to 90° but was opposite in direction.

As the model is based on the 2D Puck's theory for triggering the meso-scale damage, Mode-A failure mode was detected in the plies because the condition for this mode is that is 22 0 σ ≥  as referred in Table 2-1. The damage progression in ply 2 is shown in

Tension test with Arcan

The two laminates [0°] 16 and [0/90°] 4s were also passed through the tension loading along the fiber direction. The fractured specimen is shown in Figure 4-42 (a) where it can be seen that damage initiated at the notch root on one side and then propagated along the fibers towards the bolted holes. On the other side of the root, damage was observed on the bolted hole. In addition, the stress-strain curve is plotted and referred in Figure 4-42 (b) with the strain in x-direction and y-direction acquired by the strain gauges. The breaking stress calculated here was 123.0 MPa. The fractured specimen was examined after the test and matrix failure was witnessed rather than fiber fracture. The failure points for both of the specimens in terms of the longitudinal normal stress and transverse shear stress are plotted in Figure 4-44. θ is the angle formed between the significant section and the load. It is worth mentioning that the transverse Arcan specimen would have higher breaking strength in pure tension as compared to the crossply because cross-ply contains 50% less fiber in the loading direction. But, as no fiber fracture was obtained in the tension test of transverse ply, therefore this breaking stress is mostly associated to matrix fracture. In this section, first the cross-ply specimens were loaded in pure shear conditions in Arcan fixture up to a predefined load less than the ultimate shear strength in order to induce damage in the specimens. Then, the same specimens were unmounted and rotated at an angle of 90° in the fixture to get pure tension stress. Specimens failed at the significant section in the tensile load. The tensile behavior of the specimens having prior damage in shear are compared with the tensile stress-strain response with the Arcan specimens tested previously in tension having no damage as a priori.

The shear stress-strain profiles of the two specimens which were unloaded at 6 kN and 7 kN in pure shear stress condition in Arcan are shown in Figure 4-45. It seems that the shear stress-strain profiles applied in the first step follow each other closely in addition to the pure shear test curve of cross-ply. In the second step, the specimens were unmounted and examined carefully in order to search any macro-damage or failure at the bolted holes. No macro-damage was found and the bolted holes were also found intact. The specimens were then clamped again in the Arcan fixture where the load is in the fiber direction. Tensile load at the rate of 2 mm/min was applied until the rupture of the specimens.

Conclusion

In the first phase of the experimentation program, specimens were fabricated for the UD characterization in different directions according to ASTM standards. Elastic constants and strength were determined experimentally. Then the bi-axial testing of FRPs composites was realized. For this purpose, Arcan fixture was modified and the butterfly specimen was optimized to obtain a uniform shear stress distribution in the significant section that shall be enough to install strain gauges. After this study, Arcan specimens were fabricated which included two types of laminates: transverse Arcan specimens and cross-ply. Pure shear tests were carried out in displacement controlled mode for both types. The shear stress-strain response was observed almost similar for both types of the laminates. Moreover, the pure shear behavior of the cross-ply Arcan was also studied numerically. In addition, the specimens were also tested at 45° and 90° which correspond to the bi-axial stress case of stress ratio one and pure tension test, respectively. Finally, the damage coupling in cross-ply specimens were investigated. In these tests, first the specimen was loaded in shear up to a predefined load below the final rupture load. Then the same specimen was unmounted and rotated at 90° in the Arcan fixture. It was observed that shear damage has degraded the tensile behavior of the laminate. components. The potential fracture plane was determined by searching the plane of maximum Puck's theory stress exposure factor. In searching process, stress tensor is transformed iteratively from the principal materials coordinate system into the fracture plane coordinate system. After detecting the fracture, the searching is terminated and then damage is accumulated based on the stress components in the fracture plane coordinate system. The indentation and sliding phenomenon of FRPs against the metallic ball was numerically studied. A good correlation was observed by analyzing the failure mechanisms that prevail in the contact and adhesion wear of FRPs laminates against the metallic counterparts. Only UD based laminates were studied in adhesion wear in which fibers were in one direction in all the plies embedded in the laminates. As a matter of fact, such laminates are less susceptible to delamination, therefore in modeling the sliding delamination was not taken into account. Nevertheless, this 3D model shall be enhanced further with the elastoplastic constitutive law to capture the non-linearity when higher off-axis plies are embedded in the laminate.

Experimental work

In the first phase of the experimental part, the characterization of the materials was carried out which included the fiber direction, transverse direction and shear monotonic loading in compliant with ASTM standards. Cyclic tests of [±45°] 4s was also conducted to highlight the stiffness degradation. From these tests, the elastic constants and the strengths in the material principal directions were determined. Furthermore, the Open Hole Tension (OHT) tests of cross-ply [0/90°] 4s and [±45°] 4s were carried out. The strain fields were measured with the help of Digital Image Correlation (DIC) in the above experiments.

Then, the Arcan fixture was modified and manufactured from Aluminum Alloy AA6061 for the pure shear and combined shear/normal stress testing. It is necessary to optimize the Arcan butterfly specimen in order to get a uniform shear strain field in the significant section of the specimen where strain gauges shall be effectively installed.

For this purpose, parametric optimization of the specimen was carried out for the important dimensions that affect the stress and strain field. The outcome of this study was an optimized Arcan specimen. The required numbers of Arcan specimens were fabricated with the help of water jet cutting from the plate manufactured in-house. The pure shear tests were conducted first for the two types of stacking: [0°] 16 and [0/90°] 4s .

Afterwards, bi-axial and uni-axial loading were applied by rotating the specimens at 45° and 90° in the Arcan fixture respectively. Two strain gauges were used for the strain recording in addition to DIC.

In the last part of the experimentation program, the aim was to study the damage coupling induced in shear loading and in tension in FRPs. For the realization of this part, load was applied in two loading steps. In first step, the butterfly specimens in the Arcan were loaded to a predefined load value that can induce diffuse damages in pure shear condition. Then the specimens were unmounted and rotated at 90° in the second loading step to test it in pure tensile loading up to complete rupture. A clear drop in the slope of the tensile stress-strain behavior was observed by comparing the tests where the tensile load was applied in a single step until rupture and with the profile obtained from the tensile tests of the prior damaged Arcan specimens.

Future recommendations

Following are the suggestions to be carried out in future:

1. The coupled damage/plasticity model developed here shall be extended to the combined isotropic-kinematic hardening model in order to be used in the cyclic loading and compression.

2. The performance of the developed 3D model shall be assessed in the impact and bending loading. The delamination capability can also be added easily by defining the cohesive layer in between the two plies which is important for impact and bending loading. This model is elastic/damage model, therefore, it shall be transformed to elastoplastic damage law to predict the non-linearity specifically in dominant shear stress condition where a higher degree of nonlinearity is observed in FRPs.

3. Multi-axial laminates shall be studied while sliding against the metallic ball and delamination shall be incorporated.

4. Due to Puck's theory, the models here are limited to UD based FRPs. However, the model can be used for the fiber yarn in textile composites.

5. The damage coupling in the cross-ply through the modified Arcan fixture shall be further investigated in great detail. Titre de la thèse: Coupled damage/plasticity modeling for Non Crimp Fabrics (NCFs) Fiber Reinforced Polymers (FRPs) based on Puck's theory Absract (Français) Un modèle numérique basé sur la mécanique continue de l'endommagement couplé avec la plasticité a été développé pour l'étude des composites à matrice polymère renforcés de fibres continues (FRP). L'objectif était de fournir un outil de prévision de l'endommagement fiable et précis. Le critère de rupture de Puck a été utilisé pour prévoir le début de l'endommagement intralaminaire à l'échelle mésoscopique dans le cas d'un pli homogéinisé en 2D et en 3D. Une loi d'évolution de l'endommagement thermodynamiquement consistante a été définie pour l'accumulation de l'endommagement conduisant à la dégradation progressive de la rigidité du matériau. La prévision du comportement mécanique non linéaire ainsi que la rupture finale ont été validées sur des problèmes 2D (éprouvette trouée en traction) et en 3D (problème du contact frottant). Les résultats obtenus ont montré que le modèle développé était plus performant ceux disponibles dans la littérature scientifique. Dans la seconde partie de la thèse, un système d'essais multiaxiaux, appelé Arcan, a été développé afin d'étudier le comportement d'une éprouvette UD composite en traction en en cisaillement. Les essais ont permis de mettre en évidence l'influence de l'endommagement issu du cisaillement à l'aide d'éprouvettes optimisées.

Mots-clé:

Elastoplasticité, Théorie de Puck, Composites à matrice polymère et fibres continues, Mécanique continue de l'endommagement.

Absract (Anglais)

In the current research work, numerical modeling based on the coupled plasticity and Continuum Damage Mechanics (CDM) theory has been carried out for Fiber Reinforced Polymers (FRPs) composites. The objectives were to provide comparatively a more reliable and accurate numerical prediction tool for quasi-static loading. For this purpose, a relatively more efficient failure theory, Puck's theory, was chosen for triggering the intra-laminar meso-scale damage onset in an orthotropic homogenized ply in 2D and 3D. The thermodynamically consistent damage evolution law was defined to accumulate the damage leading to the subsequent stiffness degradation coupled with the isotropic hardening plasticity. The prediction of the mechanical behavior and the subsequent ultimate failure has been validated on 2D problem (specimen with a hole under tension) and 3D problem (sliding contact problem). Results have been significantly improved from the previous investigations. In the second part of the thesis, UD mechanical characterization was performed and Arcan fixture was modified for testing the FRPs in combined in-plane shear and tensile loading. The effects of damage induced in pure shear state were highlighted experimentally on the tensile behavior using the optimized butterfly Arcan specimen in which a fairly uniform shear stress field was observed. Elastoplasticity, Puck's theory, FRPs, Continuum Damage Mechanics.

Keywords:

I

  am also greatly indebted to my co-advisor Dr. Gérald Franz and other researchers and teachers of the laboratory specially Dr. Malek Habak and Mr. Hugues Tortajada for their encouragements that stimulated me to complete the thesis. I feel great pleasure in expressing my deep sense of obligations for the cordial cooperation and team work of Pei Hao (PhD Student). We had very fruitful discussions on the subjects in addition to testing and manufacturing the composites. I offer good wishes for his future endeavors! Moreover, I say thanks to my colleague Dr. Souhail El Bouami. He helped me to acclimatize according to the new cultural environment. I say special thanks to all my Pakistani friends particularly Mr. Azhar Hussain with whom I spent some of the weekends and enjoyed the Pakistani cuisines. Moreover, best wishes for Pouya Naderi in his PhD who is one of my best friends. Last but not the least, I am highly thankful to my dearest wife Huma Israr who took best care of my beloved daughter Umaima Israr in my absence in Pakistan along with her own studies which was indeed a challenging task for her.

4

 4 contrainte hors plan ne peut être ignorée. Le phénomène d'usure par adhérence dans le stratifié UD et les types de mécanismes de défaillance induits par l'usure par indentation et par adhérence lors du glissement dans les FRP ont été étudiés, où la modélisation 3D est inévitable. Les facteurs spécifiques d'exposition aux défaillances (données fournies par la théorie de Puck) ont été analysés. Une très bonne corrélation a été obtenue entre ces facteurs et le taux d'usure. Ils ont été comparés à la littérature disponible sur les taux d'usure par adhésion dans le cas des matériaux composites UD. Plusieurs configurations de glissement ont été considérées : parallèle (le glissement se produit dans la direction des fibres), antiparallèle (le glissement est perpendiculaire aux fibres) et des orientations intermédiaires. Les résultats ont montré une très bonne correspondance des résultats avec la variation de l'orientation de l'angle entre la fibre et la direction de glissement. En ce qui concerne les performances tribologiques, le glissement perpendiculaire à la fibre a entraîné des facteurs d'exposition aux endommagements plus élevés et donc un taux d'usure plus élevé. Ce modèle d'endommagement peut être également utilisé dans les problèmes de modélisation d'impact et de flexion. Le modèle doit encore être amélioré pour utiliser une loi de comportement non linéaire ainsi que la modélisation du délaminage basée sur la loi de traction-séparation. CHAPITRE Dans la première partie du chapitre, des éprouvettes ont été fabriquées pour la caractérisation des composites unidirectionnels UD dans différentes directions conformément aux normes ASTM. Les constantes élastiques et les résistances ont été déterminées expérimentalement. Ensuite, des tests bi-axiaux sur des composites de FRP ont été réalisés. À cette fin, le dispositif expérimental Arcan a été modifié et l'éprouvette papillon optimisée pour obtenir une distribution uniforme des contraintes de cisaillement dans la section utile. En effet celle-ci doit être suffisante pour installer des jauges de déformations. Après cette étude théorique et numérique, des éprouvette Arcan avec le xi design optimisé ont été fabriquées. Deux types de stratifiés ont été considérés: des éprouvettes avec des plis uniquement UD et éprouvettes avec des plis croisés.

Chapter 4 : 2 . 5 :

 425 In the first part of chapter 4, the material characterization is carried out according to the applicable ASTM standards. Afterward, Arcan experimentations were conducted for the pure shear and in-plane bi-axial loading. For realization of this part, a new Arcan fixture was designed and manufactured along with the optimization of the special butterfly specimen through Python scripting in ABAQUS environment. The motivation of the optimization was to get a fairly uniform shear domain in the significant section of the Arcan specimen where strain gauges can be effectively installed. Apart from the pure shear test and bi-axial tests, the newly designed Arcan fixture was also used to highlight the effects of the diffused damages induced in pure shear on the tensile behavior of FRPs. Furthermore, the pure shear experimental results were supplemented with the numerical simulations by employing the model developed in chapter Chapter The final chapter deals with the findings and the conclusions pertaining to the numerical and experimental domains of the present thesis. In addition, the limitations of the model were discussed and suggestions were made in future line of action.

3 1. 3 . 1

 331 étendu en français ............................................................................................ v General Introduction ....................................................................................................... xii .................................................................................................................................... Chapter 1 Introduction .................................................................................................................. 1.1 Importance of composites....................................................................................... 1.2 Composites and its classification ......................................................................... 1.Textile composites............................................................................................... Bi-angle NCFs .............................................................................................................. 1.4 1.4.1 Applications of shallow bi-angle NCFs ....................................................... 1.4.2 Manufacturing of bi-angle tapes ................................................................... Failure mechanisms in FRPs ................................................................................. 1.5 1.5.1 Multi-scale nature of the composite structure ........................................ Failure theories for FRPs ...................................................................................... 1.6 Damage modeling in FRPs ................................................................................... 1.7 1.7.1 Progressive damage modeling ..................................................................... 1.7.2 Damage mechanics and phenomenological approach ........................ 1.7.3 Meso-scale damage modeling ...................................................................... Plasticity and damage coupling in FRPs ......................................................... 1.8 1.8.1 Plasticity and damage potentials ................................................................ 1.8.2 Return Mapping Algorithm (RMA) ............................................................. .................................................................................................................................. Chapter 2 Introduction ................................................................................................................ 2.1 Damage/plasticity coupled material modeling ............................................. 2.2 2.2.1 Damage/plasticity coupled meso-models ................................................. 2.2.2 Anisotropic plasticity damage coupled constitutive law.................... 2.2.3 Meso-damage onset functions ...................................................................... 2.2.4 Damage accumulation functions ................................................................. 2.2.5 Phenomenological plasticity modeling ..................................................... 2.2.6 Consistent algorithm operator ..................................................................... FEM implementation of the proposed model ................................................

Figure 1 - 1 :

 11 Figure1-1: Materials distribution in Boeing B-787[START_REF] Campbell | Manufacturing Technology for Aerospace Structural Materials[END_REF] ........................................................ Figure1-2 Percentages of the composites in military and civilian aircraft structures[START_REF] Kassapoglou | Design and analysis of composite structures: with application to aerospace structures[END_REF] Figure1-3 Global automotive consumption of carbon fiber composite (thousands of tons) ....................... Figure1-4: Few examples of the textile composite types[START_REF] Lomov Sv | Textile Composite Materials: Polymer Matrix Composites[END_REF] ............................................. Figure 1-5: (a) Fabrication of Quadri-axial NCF preforms , (b) Front and back fabrics of a ±45° preform (Mattsson D, 2007) ...................................................................................................................................... Figure 1-6: (a) Ultra-thin plies laminates used in solar powered aircraft (Amacher R, 2014), (b) applications of bi-angle thin plies based NCFs in slender structures (Massard T, 2011) ............................ Figure 1-7: Pneumatic compression to convert thick tow into thin tow (Sihn S, 2007) .............................. Figure 1-8: (a) CF12K tow before spreading (width= 6 mm), (b) spread tow (width= 12 mm), (c) spread tow (width= 20 mm) (Sihn S, 2007) ............................................................................................................Figure 1-9 (a) Shallow bi-angle fabrication process (b) commercially available bi-angle NCFs fabric having 0.125 mm thickness (Tsai, 2011) ..................................................................................................... Figure 1-10: Multi-scale nature of the composite structure ......................................................................... Figure 1-11: Schematics of the failure mechanisms in UD FRPs with circular hole under tensile load (van der Meer, 2012) ................................................................................................................................. Figure 1-12: Different failure mechanisms in UD FRPs (Ramesh and Chandra, 2012) ............................ Figure 1-13: Flow chart of a typical progressive damage model and scale-linking (Liu, 2010) ................ Figure 1-14: Global load-displacement plot of ABAQUS built-in model for quasi-isotropic laminate (Ulrich et al., 2015) .................................................................................................................................... Figure 1-15: Types of Inter Fiber Fractures (IFFs) and stresses (Markus, 2016) ...................................... Figure 1-16: (a) Progressive damage accumulation proportional to the applied loading , (b) IFFs in the CFRP and Glass Fiber Reinforced Polymer (GFRP) tubes ........................................................................ Figure 1-17: Three modes of fracture in Puck's theory .............................................................................. Figure 1-18: Wedge effect in Mode-C (Knops, 2008) ............................................................................... Figure 1-19: Inclined fracture in compression (Mode-C) .......................................................................... Figure 1-20: Geometrical representation of Return Mapping Algorithim (RMA) (Simo JC, 1985)..........Figure 2-1: FEM Implementation flow chart ............................................................................................. Figure 2-2: Damage growth and its dependence on the mesh size ............................................................ Figure 2-3: (a) Mesh sensitivity in fiber direction, (b) mesh sensitivity in transverse direction, (c) mesh sensitivity in shear, (d) bi-angle longitudinal stress-strain prediction ........................................................ Figure 2-4: Quarter symmetric OHT geometry with Boundary Conditions (BCs) .................................... Figure 2-5: Longitudinal stress-strain response of laminate [(0/±45/90) 3 ] s and comparison with the model developed by Zhang et al. (Zhang, 2011). ................................................................................................. Figure 2-6: Present model and ABAQUS built-in elastic/damage model for OHT [(±45) 6 ] s .................... Figure 2-7: Transverse crack growth and local delamination after the saturation crack density in [0/90°] s laminate (Paris et al., 2010) ......................................................................................................................Figure 2-8: Matrix damage accumulation in 90° ply at (a) 12800 N, (b) 20281 N (c) ultimate load, 25394 N ................................................................................................................................................................Figure 2-9: Fiber damage progression in the 0° ply at (a) 19169 N, (b) 22263 N (c) ultimate load, 25394 N ................................................................................................................................................................Figure 2-10: Present model in comparison with Lee et al. (Lee et al., 2015) ............................................Figure 3-1: (a) Plane stress case, (b) transition from plane stress into 3D state of stress ..........................Figure 3-2: Stress components in 1-t-n and 1-2-3 coordinate system (Puck et al., 2002) .........................Figure 3-3: Fracture planes of the transverse cracks (IFF) corresponding to the type of stress (Matthias Deuschle, 2012) .........................................................................................................................................Figure 3-4: Weakening provoking range ...................................................................................................Figure 3-5: Shrinkage of IFF failure surface due to FF: Puck's Cigar .......................................................

Figure 2 -

 2 Figure1-1: Materials distribution in Boeing B-787[START_REF] Campbell | Manufacturing Technology for Aerospace Structural Materials[END_REF] ........................................................ Figure1-2 Percentages of the composites in military and civilian aircraft structures[START_REF] Kassapoglou | Design and analysis of composite structures: with application to aerospace structures[END_REF] Figure1-3 Global automotive consumption of carbon fiber composite (thousands of tons) ....................... Figure1-4: Few examples of the textile composite types[START_REF] Lomov Sv | Textile Composite Materials: Polymer Matrix Composites[END_REF] ............................................. Figure 1-5: (a) Fabrication of Quadri-axial NCF preforms , (b) Front and back fabrics of a ±45° preform (Mattsson D, 2007) ...................................................................................................................................... Figure 1-6: (a) Ultra-thin plies laminates used in solar powered aircraft (Amacher R, 2014), (b) applications of bi-angle thin plies based NCFs in slender structures (Massard T, 2011) ............................ Figure 1-7: Pneumatic compression to convert thick tow into thin tow (Sihn S, 2007) .............................. Figure 1-8: (a) CF12K tow before spreading (width= 6 mm), (b) spread tow (width= 12 mm), (c) spread tow (width= 20 mm) (Sihn S, 2007) ............................................................................................................Figure 1-9 (a) Shallow bi-angle fabrication process (b) commercially available bi-angle NCFs fabric having 0.125 mm thickness (Tsai, 2011) ..................................................................................................... Figure 1-10: Multi-scale nature of the composite structure ......................................................................... Figure 1-11: Schematics of the failure mechanisms in UD FRPs with circular hole under tensile load (van der Meer, 2012) ................................................................................................................................. Figure 1-12: Different failure mechanisms in UD FRPs (Ramesh and Chandra, 2012) ............................ Figure 1-13: Flow chart of a typical progressive damage model and scale-linking (Liu, 2010) ................ Figure 1-14: Global load-displacement plot of ABAQUS built-in model for quasi-isotropic laminate (Ulrich et al., 2015) .................................................................................................................................... Figure 1-15: Types of Inter Fiber Fractures (IFFs) and stresses (Markus, 2016) ...................................... Figure 1-16: (a) Progressive damage accumulation proportional to the applied loading , (b) IFFs in the CFRP and Glass Fiber Reinforced Polymer (GFRP) tubes ........................................................................ Figure 1-17: Three modes of fracture in Puck's theory .............................................................................. Figure 1-18: Wedge effect in Mode-C (Knops, 2008) ............................................................................... Figure 1-19: Inclined fracture in compression (Mode-C) .......................................................................... Figure 1-20: Geometrical representation of Return Mapping Algorithim (RMA) (Simo JC, 1985)..........Figure 2-1: FEM Implementation flow chart ............................................................................................. Figure 2-2: Damage growth and its dependence on the mesh size ............................................................ Figure 2-3: (a) Mesh sensitivity in fiber direction, (b) mesh sensitivity in transverse direction, (c) mesh sensitivity in shear, (d) bi-angle longitudinal stress-strain prediction ........................................................ Figure 2-4: Quarter symmetric OHT geometry with Boundary Conditions (BCs) .................................... Figure 2-5: Longitudinal stress-strain response of laminate [(0/±45/90) 3 ] s and comparison with the model developed by Zhang et al. (Zhang, 2011). ................................................................................................. Figure 2-6: Present model and ABAQUS built-in elastic/damage model for OHT [(±45) 6 ] s .................... Figure 2-7: Transverse crack growth and local delamination after the saturation crack density in [0/90°] s laminate (Paris et al., 2010) ......................................................................................................................Figure 2-8: Matrix damage accumulation in 90° ply at (a) 12800 N, (b) 20281 N (c) ultimate load, 25394 N ................................................................................................................................................................Figure 2-9: Fiber damage progression in the 0° ply at (a) 19169 N, (b) 22263 N (c) ultimate load, 25394 N ................................................................................................................................................................Figure 2-10: Present model in comparison with Lee et al. (Lee et al., 2015) ............................................Figure 3-1: (a) Plane stress case, (b) transition from plane stress into 3D state of stress ..........................Figure 3-2: Stress components in 1-t-n and 1-2-3 coordinate system (Puck et al., 2002) .........................Figure 3-3: Fracture planes of the transverse cracks (IFF) corresponding to the type of stress (Matthias Deuschle, 2012) .........................................................................................................................................Figure 3-4: Weakening provoking range ...................................................................................................Figure 3-5: Shrinkage of IFF failure surface due to FF: Puck's Cigar .......................................................

Figure 2 - 9 :

 29 Figure1-1: Materials distribution in Boeing B-787[START_REF] Campbell | Manufacturing Technology for Aerospace Structural Materials[END_REF] ........................................................ Figure1-2 Percentages of the composites in military and civilian aircraft structures[START_REF] Kassapoglou | Design and analysis of composite structures: with application to aerospace structures[END_REF] Figure1-3 Global automotive consumption of carbon fiber composite (thousands of tons) ....................... Figure1-4: Few examples of the textile composite types[START_REF] Lomov Sv | Textile Composite Materials: Polymer Matrix Composites[END_REF] ............................................. Figure 1-5: (a) Fabrication of Quadri-axial NCF preforms , (b) Front and back fabrics of a ±45° preform (Mattsson D, 2007) ...................................................................................................................................... Figure 1-6: (a) Ultra-thin plies laminates used in solar powered aircraft (Amacher R, 2014), (b) applications of bi-angle thin plies based NCFs in slender structures (Massard T, 2011) ............................ Figure 1-7: Pneumatic compression to convert thick tow into thin tow (Sihn S, 2007) .............................. Figure 1-8: (a) CF12K tow before spreading (width= 6 mm), (b) spread tow (width= 12 mm), (c) spread tow (width= 20 mm) (Sihn S, 2007) ............................................................................................................Figure 1-9 (a) Shallow bi-angle fabrication process (b) commercially available bi-angle NCFs fabric having 0.125 mm thickness (Tsai, 2011) ..................................................................................................... Figure 1-10: Multi-scale nature of the composite structure ......................................................................... Figure 1-11: Schematics of the failure mechanisms in UD FRPs with circular hole under tensile load (van der Meer, 2012) ................................................................................................................................. Figure 1-12: Different failure mechanisms in UD FRPs (Ramesh and Chandra, 2012) ............................ Figure 1-13: Flow chart of a typical progressive damage model and scale-linking (Liu, 2010) ................ Figure 1-14: Global load-displacement plot of ABAQUS built-in model for quasi-isotropic laminate (Ulrich et al., 2015) .................................................................................................................................... Figure 1-15: Types of Inter Fiber Fractures (IFFs) and stresses (Markus, 2016) ...................................... Figure 1-16: (a) Progressive damage accumulation proportional to the applied loading , (b) IFFs in the CFRP and Glass Fiber Reinforced Polymer (GFRP) tubes ........................................................................ Figure 1-17: Three modes of fracture in Puck's theory .............................................................................. Figure 1-18: Wedge effect in Mode-C (Knops, 2008) ............................................................................... Figure 1-19: Inclined fracture in compression (Mode-C) .......................................................................... Figure 1-20: Geometrical representation of Return Mapping Algorithim (RMA) (Simo JC, 1985)..........Figure 2-1: FEM Implementation flow chart ............................................................................................. Figure 2-2: Damage growth and its dependence on the mesh size ............................................................ Figure 2-3: (a) Mesh sensitivity in fiber direction, (b) mesh sensitivity in transverse direction, (c) mesh sensitivity in shear, (d) bi-angle longitudinal stress-strain prediction ........................................................ Figure 2-4: Quarter symmetric OHT geometry with Boundary Conditions (BCs) .................................... Figure 2-5: Longitudinal stress-strain response of laminate [(0/±45/90) 3 ] s and comparison with the model developed by Zhang et al. (Zhang, 2011). ................................................................................................. Figure 2-6: Present model and ABAQUS built-in elastic/damage model for OHT [(±45) 6 ] s .................... Figure 2-7: Transverse crack growth and local delamination after the saturation crack density in [0/90°] s laminate (Paris et al., 2010) ......................................................................................................................Figure 2-8: Matrix damage accumulation in 90° ply at (a) 12800 N, (b) 20281 N (c) ultimate load, 25394 N ................................................................................................................................................................Figure 2-9: Fiber damage progression in the 0° ply at (a) 19169 N, (b) 22263 N (c) ultimate load, 25394 N ................................................................................................................................................................Figure 2-10: Present model in comparison with Lee et al. (Lee et al., 2015) ............................................Figure 3-1: (a) Plane stress case, (b) transition from plane stress into 3D state of stress ..........................Figure 3-2: Stress components in 1-t-n and 1-2-3 coordinate system (Puck et al., 2002) .........................Figure 3-3: Fracture planes of the transverse cracks (IFF) corresponding to the type of stress (Matthias Deuschle, 2012) .........................................................................................................................................Figure 3-4: Weakening provoking range ...................................................................................................Figure 3-5: Shrinkage of IFF failure surface due to FF: Puck's Cigar .......................................................

Figure 3 -

 3 Figure1-1: Materials distribution in Boeing B-787[START_REF] Campbell | Manufacturing Technology for Aerospace Structural Materials[END_REF] ........................................................ Figure1-2 Percentages of the composites in military and civilian aircraft structures[START_REF] Kassapoglou | Design and analysis of composite structures: with application to aerospace structures[END_REF] Figure1-3 Global automotive consumption of carbon fiber composite (thousands of tons) ....................... Figure1-4: Few examples of the textile composite types[START_REF] Lomov Sv | Textile Composite Materials: Polymer Matrix Composites[END_REF] ............................................. Figure 1-5: (a) Fabrication of Quadri-axial NCF preforms , (b) Front and back fabrics of a ±45° preform (Mattsson D, 2007) ...................................................................................................................................... Figure 1-6: (a) Ultra-thin plies laminates used in solar powered aircraft (Amacher R, 2014), (b) applications of bi-angle thin plies based NCFs in slender structures (Massard T, 2011) ............................ Figure 1-7: Pneumatic compression to convert thick tow into thin tow (Sihn S, 2007) .............................. Figure 1-8: (a) CF12K tow before spreading (width= 6 mm), (b) spread tow (width= 12 mm), (c) spread tow (width= 20 mm) (Sihn S, 2007) ............................................................................................................Figure 1-9 (a) Shallow bi-angle fabrication process (b) commercially available bi-angle NCFs fabric having 0.125 mm thickness (Tsai, 2011) ..................................................................................................... Figure 1-10: Multi-scale nature of the composite structure ......................................................................... Figure 1-11: Schematics of the failure mechanisms in UD FRPs with circular hole under tensile load (van der Meer, 2012) ................................................................................................................................. Figure 1-12: Different failure mechanisms in UD FRPs (Ramesh and Chandra, 2012) ............................ Figure 1-13: Flow chart of a typical progressive damage model and scale-linking (Liu, 2010) ................ Figure 1-14: Global load-displacement plot of ABAQUS built-in model for quasi-isotropic laminate (Ulrich et al., 2015) .................................................................................................................................... Figure 1-15: Types of Inter Fiber Fractures (IFFs) and stresses (Markus, 2016) ...................................... Figure 1-16: (a) Progressive damage accumulation proportional to the applied loading , (b) IFFs in the CFRP and Glass Fiber Reinforced Polymer (GFRP) tubes ........................................................................ Figure 1-17: Three modes of fracture in Puck's theory .............................................................................. Figure 1-18: Wedge effect in Mode-C (Knops, 2008) ............................................................................... Figure 1-19: Inclined fracture in compression (Mode-C) .......................................................................... Figure 1-20: Geometrical representation of Return Mapping Algorithim (RMA) (Simo JC, 1985)..........Figure 2-1: FEM Implementation flow chart ............................................................................................. Figure 2-2: Damage growth and its dependence on the mesh size ............................................................ Figure 2-3: (a) Mesh sensitivity in fiber direction, (b) mesh sensitivity in transverse direction, (c) mesh sensitivity in shear, (d) bi-angle longitudinal stress-strain prediction ........................................................ Figure 2-4: Quarter symmetric OHT geometry with Boundary Conditions (BCs) .................................... Figure 2-5: Longitudinal stress-strain response of laminate [(0/±45/90) 3 ] s and comparison with the model developed by Zhang et al. (Zhang, 2011). ................................................................................................. Figure 2-6: Present model and ABAQUS built-in elastic/damage model for OHT [(±45) 6 ] s .................... Figure 2-7: Transverse crack growth and local delamination after the saturation crack density in [0/90°] s laminate (Paris et al., 2010) ......................................................................................................................Figure 2-8: Matrix damage accumulation in 90° ply at (a) 12800 N, (b) 20281 N (c) ultimate load, 25394 N ................................................................................................................................................................Figure 2-9: Fiber damage progression in the 0° ply at (a) 19169 N, (b) 22263 N (c) ultimate load, 25394 N ................................................................................................................................................................Figure 2-10: Present model in comparison with Lee et al. (Lee et al., 2015) ............................................Figure 3-1: (a) Plane stress case, (b) transition from plane stress into 3D state of stress ..........................Figure 3-2: Stress components in 1-t-n and 1-2-3 coordinate system (Puck et al., 2002) .........................Figure 3-3: Fracture planes of the transverse cracks (IFF) corresponding to the type of stress (Matthias Deuschle, 2012) .........................................................................................................................................Figure 3-4: Weakening provoking range ...................................................................................................Figure 3-5: Shrinkage of IFF failure surface due to FF: Puck's Cigar .......................................................

Figure 3 - 2 :

 32 Figure1-1: Materials distribution in Boeing B-787[START_REF] Campbell | Manufacturing Technology for Aerospace Structural Materials[END_REF] ........................................................ Figure1-2 Percentages of the composites in military and civilian aircraft structures[START_REF] Kassapoglou | Design and analysis of composite structures: with application to aerospace structures[END_REF] Figure1-3 Global automotive consumption of carbon fiber composite (thousands of tons) ....................... Figure1-4: Few examples of the textile composite types[START_REF] Lomov Sv | Textile Composite Materials: Polymer Matrix Composites[END_REF] ............................................. Figure 1-5: (a) Fabrication of Quadri-axial NCF preforms , (b) Front and back fabrics of a ±45° preform (Mattsson D, 2007) ...................................................................................................................................... Figure 1-6: (a) Ultra-thin plies laminates used in solar powered aircraft (Amacher R, 2014), (b) applications of bi-angle thin plies based NCFs in slender structures (Massard T, 2011) ............................ Figure 1-7: Pneumatic compression to convert thick tow into thin tow (Sihn S, 2007) .............................. Figure 1-8: (a) CF12K tow before spreading (width= 6 mm), (b) spread tow (width= 12 mm), (c) spread tow (width= 20 mm) (Sihn S, 2007) ............................................................................................................Figure 1-9 (a) Shallow bi-angle fabrication process (b) commercially available bi-angle NCFs fabric having 0.125 mm thickness (Tsai, 2011) ..................................................................................................... Figure 1-10: Multi-scale nature of the composite structure ......................................................................... Figure 1-11: Schematics of the failure mechanisms in UD FRPs with circular hole under tensile load (van der Meer, 2012) ................................................................................................................................. Figure 1-12: Different failure mechanisms in UD FRPs (Ramesh and Chandra, 2012) ............................ Figure 1-13: Flow chart of a typical progressive damage model and scale-linking (Liu, 2010) ................ Figure 1-14: Global load-displacement plot of ABAQUS built-in model for quasi-isotropic laminate (Ulrich et al., 2015) .................................................................................................................................... Figure 1-15: Types of Inter Fiber Fractures (IFFs) and stresses (Markus, 2016) ...................................... Figure 1-16: (a) Progressive damage accumulation proportional to the applied loading , (b) IFFs in the CFRP and Glass Fiber Reinforced Polymer (GFRP) tubes ........................................................................ Figure 1-17: Three modes of fracture in Puck's theory .............................................................................. Figure 1-18: Wedge effect in Mode-C (Knops, 2008) ............................................................................... Figure 1-19: Inclined fracture in compression (Mode-C) .......................................................................... Figure 1-20: Geometrical representation of Return Mapping Algorithim (RMA) (Simo JC, 1985)..........Figure 2-1: FEM Implementation flow chart ............................................................................................. Figure 2-2: Damage growth and its dependence on the mesh size ............................................................ Figure 2-3: (a) Mesh sensitivity in fiber direction, (b) mesh sensitivity in transverse direction, (c) mesh sensitivity in shear, (d) bi-angle longitudinal stress-strain prediction ........................................................ Figure 2-4: Quarter symmetric OHT geometry with Boundary Conditions (BCs) .................................... Figure 2-5: Longitudinal stress-strain response of laminate [(0/±45/90) 3 ] s and comparison with the model developed by Zhang et al. (Zhang, 2011). ................................................................................................. Figure 2-6: Present model and ABAQUS built-in elastic/damage model for OHT [(±45) 6 ] s .................... Figure 2-7: Transverse crack growth and local delamination after the saturation crack density in [0/90°] s laminate (Paris et al., 2010) ......................................................................................................................Figure 2-8: Matrix damage accumulation in 90° ply at (a) 12800 N, (b) 20281 N (c) ultimate load, 25394 N ................................................................................................................................................................Figure 2-9: Fiber damage progression in the 0° ply at (a) 19169 N, (b) 22263 N (c) ultimate load, 25394 N ................................................................................................................................................................Figure 2-10: Present model in comparison with Lee et al. (Lee et al., 2015) ............................................Figure 3-1: (a) Plane stress case, (b) transition from plane stress into 3D state of stress ..........................Figure 3-2: Stress components in 1-t-n and 1-2-3 coordinate system (Puck et al., 2002) .........................Figure 3-3: Fracture planes of the transverse cracks (IFF) corresponding to the type of stress (Matthias Deuschle, 2012) .........................................................................................................................................Figure 3-4: Weakening provoking range ...................................................................................................Figure 3-5: Shrinkage of IFF failure surface due to FF: Puck's Cigar .......................................................

Figure 3 - 3 :Figure 3 - 6 :Figure 3 - 9 :Figure 3 -Figure 4 - 1 :Figure 4 - 8 :Figure 4 - 9 :

 3336393414849 Figure1-1: Materials distribution in Boeing B-787[START_REF] Campbell | Manufacturing Technology for Aerospace Structural Materials[END_REF] ........................................................ Figure1-2 Percentages of the composites in military and civilian aircraft structures[START_REF] Kassapoglou | Design and analysis of composite structures: with application to aerospace structures[END_REF] Figure1-3 Global automotive consumption of carbon fiber composite (thousands of tons) ....................... Figure1-4: Few examples of the textile composite types[START_REF] Lomov Sv | Textile Composite Materials: Polymer Matrix Composites[END_REF] ............................................. Figure 1-5: (a) Fabrication of Quadri-axial NCF preforms , (b) Front and back fabrics of a ±45° preform (Mattsson D, 2007) ...................................................................................................................................... Figure 1-6: (a) Ultra-thin plies laminates used in solar powered aircraft (Amacher R, 2014), (b) applications of bi-angle thin plies based NCFs in slender structures (Massard T, 2011) ............................ Figure 1-7: Pneumatic compression to convert thick tow into thin tow (Sihn S, 2007) .............................. Figure 1-8: (a) CF12K tow before spreading (width= 6 mm), (b) spread tow (width= 12 mm), (c) spread tow (width= 20 mm) (Sihn S, 2007) ............................................................................................................Figure 1-9 (a) Shallow bi-angle fabrication process (b) commercially available bi-angle NCFs fabric having 0.125 mm thickness (Tsai, 2011) ..................................................................................................... Figure 1-10: Multi-scale nature of the composite structure ......................................................................... Figure 1-11: Schematics of the failure mechanisms in UD FRPs with circular hole under tensile load (van der Meer, 2012) ................................................................................................................................. Figure 1-12: Different failure mechanisms in UD FRPs (Ramesh and Chandra, 2012) ............................ Figure 1-13: Flow chart of a typical progressive damage model and scale-linking (Liu, 2010) ................ Figure 1-14: Global load-displacement plot of ABAQUS built-in model for quasi-isotropic laminate (Ulrich et al., 2015) .................................................................................................................................... Figure 1-15: Types of Inter Fiber Fractures (IFFs) and stresses (Markus, 2016) ...................................... Figure 1-16: (a) Progressive damage accumulation proportional to the applied loading , (b) IFFs in the CFRP and Glass Fiber Reinforced Polymer (GFRP) tubes ........................................................................ Figure 1-17: Three modes of fracture in Puck's theory .............................................................................. Figure 1-18: Wedge effect in Mode-C (Knops, 2008) ............................................................................... Figure 1-19: Inclined fracture in compression (Mode-C) .......................................................................... Figure 1-20: Geometrical representation of Return Mapping Algorithim (RMA) (Simo JC, 1985)..........Figure 2-1: FEM Implementation flow chart ............................................................................................. Figure 2-2: Damage growth and its dependence on the mesh size ............................................................ Figure 2-3: (a) Mesh sensitivity in fiber direction, (b) mesh sensitivity in transverse direction, (c) mesh sensitivity in shear, (d) bi-angle longitudinal stress-strain prediction ........................................................ Figure 2-4: Quarter symmetric OHT geometry with Boundary Conditions (BCs) .................................... Figure 2-5: Longitudinal stress-strain response of laminate [(0/±45/90) 3 ] s and comparison with the model developed by Zhang et al. (Zhang, 2011). ................................................................................................. Figure 2-6: Present model and ABAQUS built-in elastic/damage model for OHT [(±45) 6 ] s .................... Figure 2-7: Transverse crack growth and local delamination after the saturation crack density in [0/90°] s laminate (Paris et al., 2010) ......................................................................................................................Figure 2-8: Matrix damage accumulation in 90° ply at (a) 12800 N, (b) 20281 N (c) ultimate load, 25394 N ................................................................................................................................................................Figure 2-9: Fiber damage progression in the 0° ply at (a) 19169 N, (b) 22263 N (c) ultimate load, 25394 N ................................................................................................................................................................Figure 2-10: Present model in comparison with Lee et al. (Lee et al., 2015) ............................................Figure 3-1: (a) Plane stress case, (b) transition from plane stress into 3D state of stress ..........................Figure 3-2: Stress components in 1-t-n and 1-2-3 coordinate system (Puck et al., 2002) .........................Figure 3-3: Fracture planes of the transverse cracks (IFF) corresponding to the type of stress (Matthias Deuschle, 2012) .........................................................................................................................................Figure 3-4: Weakening provoking range ...................................................................................................Figure 3-5: Shrinkage of IFF failure surface due to FF: Puck's Cigar .......................................................

Figure 4 -

 4 Figure 4-11: In-plane shear stress-strain profile obtained in tension test of [±45°] 2s .................................

Figure 4 -

 4 Figure 4-12: (a) Fractured specimen No.4, (b) percentage normal strain field in load direction ( ) yy ε ......

Figure 4 -

 4 Figure 4-13: Loading-unloading curve ....................................................................................................Figure 4-14: Longitudinal stress xxσ and the strains in x and y directions of [±45°] 4sOHT ....................

Figure 4 -

 4 Figure 4-15: [±45°] 4s OHT tension test, (a) Fractured sample, (b)

Figure 4 -

 4 Figure 4-16: [0/90°] 4s OHT tension testing, (a) Fractured sample, (b)

Figure 4 -

 4 Figure 4-19: Butterfly-shaped Arcan Specimens under (a) tension load, P

Figure 4 -

 4 Figure 4-20: Backside of the newly designed Arcan fixture, (a) Guide shaft locking position, (b) linear ball bearings used for guide shaft support ............................................................................................... Figure 4-21: (a) Yen modified Arcan (Yen et al., 1998), (b) Present modified Arcan ............................Figure 4-22: Modified Arcan integration with the in-house universal testing machine ...........................Figure 4-23: (a) Layout of optimized Arcan specimen for UD laminate , (b) assessment of the normalized shear stress distribution, (c) Pure shear stress-strain using the gauge rosettes (El-Hajjar, 2004) ............Figure 4-24: (a) Load and BCs, (b) FEM meshing, (c) [0°] 16 UD Arcan (El-Hajjar et al. notch parameters), (d) optimized Arcan for cross ply laminate [0/90°] 4s .......................................................... Figure 4-25: (a) Optimized Arcan specimen for [0/90°] 2s , (b) Pure shear test set-up .............................. Figure 4-26: Modified Arcan specimen (a) Notch angle effect on the stress field, (b) present Arcan sample for the two different laminates ................................................................................................................. Figure 4-27: Shear stress-strain response from pure shear test (El-Hajjar, 2004) .................................... Figure 4-28: Two Arcan configurations, [0°] 16 and [0/90°] 4s .................................................................. Figure 4-29: Experimental set-up ............................................................................................................ Figure 4-30: Stress components induced internally as a function of the fiber angle when a uniaxial external stress xx σ is applied .....................................................................................................................

Figure 1 - 2

 12 Figure 1-2 Percentages of the composites in military and civilian aircraft structures[START_REF] Kassapoglou | Design and analysis of composite structures: with application to aerospace structures[END_REF]

Figure 1 - 4 :

 14 Figure 1-4: Few examples of the textile composite types (Lomov SV, 2010) 1.3.1.1 Non Crimp Fabrics (NCFs)

Figure 1

 1 Figure 1-5: (a) Fabrication of Quadri-axial NCF preforms , (b) Front and back fabrics of a ±45° preform (Mattsson D, 2007)

Figure 1

 1 Figure 1-6: (a) Ultra-thin plies laminates used in solar powered aircraft (Amacher R, 2014), (b) applications of bi-angle thin plies based NCFs in slender structures (Massard T, 2011)

Figure 1 - 7 :

 17 Figure 1-7: Pneumatic compression to convert thick tow into thin tow (Sihn S, 2007)

Figure 1 - 9 :

 19 Figure 1-9: (a) Shallow bi-angle fabrication process (b) commercially available bi-angle NCFs fabric having 0.125 mm thickness[START_REF] Tsai | Weight and cost reduction by using unbalanced and un-symmetric lamiantes[END_REF]

Figure 1 - 10 :

 110 Figure 1-10: Multi-scale nature of the composite structure

Figure 1 -

 1 Figure 1-11: Schematics of the failure mechanisms in UD FRPs with circular hole under tensile load (van der Meer, 2012)

Figure 1 -

 1 Figure 1-13: Flow chart of a typical progressive damage model and scale-linking[START_REF] Liu | Recent developments on damage modeling and finite element analysis for composite laminates: A review[END_REF]

Figure 1 -

 1 Figure1-14: Global load-displacement plot of ABAQUS built-in model for quasi-isotropic laminate[START_REF] Ulrich | Laminate damage model for composite structures[END_REF]

Figure 1 -

 1 Figure1-15: Types of Inter Fiber Fractures (IFFs) and stresses[START_REF] Markus | Failure of Fiber-Reinforced Composites[END_REF]

Figure 1 -

 1 Figure 1-16: (a) Progressive damage accumulation proportional to the applied loading , (b) IFFs in the CFRP and Glass Fiber Reinforced Polymer (GFRP) tubes Puck et al. assumed that the tensile transverse stress promoted the cracks and the transverse compressive stress impedes or slow down the cracks (Puck et al., 2002, Puck, 2002b). Consequently, different expressions for the transverse tensile and transverse compressive stress were formularized which result into asymmetrical failure envelope shown in Figure 1-17. Moreover, two different expressions for the fiber direction in tension and compression were also formed. Numerous experimental studies showed that a moderate transverse compression increases the apparent shear strength of the materials due to the increase in the friction between the sliding surfaces (Mohr-Coulomb hypothesis). This is the reason why the transverse compression was further categorized

Figure 1 - 17 :

 117 Figure 1-17: Three modes of fracture in Puck's theory Mode-A describes the mode of failure when a transverse tensile stress is applied as illustrated in Figure 1-17.The fracture plane having angle at least ±30° in Mode-C will hit the adjacent lamina with sharp edges and this is a very dangerous phenomena known as wedge effect. It can cause to explode the laminate by promoting delamination, or can induce local buckling (Knops, 2008) as depicted in Figure 1-18. Hence, Mode-C is not tolerable damage mode.

Figure 1 -

 1 Figure1-18: Wedge effect in Mode-C[START_REF] Knops | Analysis of Failure in Fiber Polymer Laminates: The Theory of Alfred Puck[END_REF]

Figure 1 -

 1 Figure 1-19: Inclined fracture in compression (Mode-C)

  Here 1 d , 2 d are the scalar damage variables in the material principal axis, and 21 d is the inplane shear damage variable. These damage state variables must be monotonically increased due to the irreversibility of the damage proportional to the loading in order to satisfy the Clausius-Duhem in-equality of the positive dissipation. As the behavior of the composite materials is unsymmetrical for the loading in tension and compression therefore different damage variables shall be defined in tension and compression. Superscripts 'T' and 'C' denote tension and compression. The damage variables are denoted as: 1 1 1

  Delamination is not considered in the present 2D model. Here, two failure exposure factors are used for the FFs in tension and compression: have been devised byPuck et al. for Inter Fiber Fracture (IFFs) initiation: transverse tension (Mode A) denoted by , ,

  represent the strengths in the fiber direction, transverse direction and in-plane shear for UD ply. c

  known from the previous increment. The trial variables are expressed based on the elastic constitutive law:

  but in case of quadratic plasticity function, these two quantities are not equal. The effective equivalent plastic deformation increment dp  can be obtained by equating the plastic work produced by the equivalent stress,

  ).All the preceding mathematical expressions are sufficient to implement the elastoplastic CDM model in ABAQUS/Standard through UMAT except the requirement of the elastoplastic consistent tangent matrix (Consistent Jacobian:

  convergence. It represents the change in the i th stress component at the end of the time increment caused by an infinitesimal perturbation of the j th component of the strain tensor:

Figure 2

 2 Figure 2-1: FEM Implementation flow chart

Figure 2 - 2 :

 22 Figure 2-2: Damage growth and its dependence on the mesh size

  are the strengths of the homogenized orthotropic UD ply. The energy release rates are measured using precracked specimens following the Linear Elastic Fracture Mechanics (LEFM). The tensile fracture energy in the transverse direction 22 T  is measured using Double

Figure 2 - 3 :

 23 Figure 2-3: (a) Mesh sensitivity in fiber direction, (b) mesh sensitivity in transverse direction, (c) mesh sensitivity in shear, (d) bi-angle longitudinal stress-strain prediction

Figure 2 - 4 (

 24 [START_REF] Vasiukov | Non-linear material modeling of fiber-reinforced polymers based on coupled viscoelasticity-viscoplasticity with anisotropic continuous damage mechanics[END_REF]: The thickness of each ply is 0.1308 mm for the three laminate configurations where total of 20 plies are embedded: [0/(±45) 3 /90 3 ] s , [0/(±45) 2 /90 5 ] s , [0/(±45)/90 7 ] s . In addition, the laminates [(0/±45/90) 3 ] s , [(±45) 6 ] s and [(0/90) 6 ] s are also analyzed comprising of 24 plies each which result into per ply thickness of 0.1322 mm[START_REF] Lee | Initial and progressive failure analyses for composite laminates using Puck failure criterion and damage-coupled finite element method[END_REF], and 0.109 mm in

Figure 2 - 4 :

 24 Figure 2-4: Quarter symmetric OHT geometry with Boundary Conditions (BCs)

  is different than the meso-model in the present case as well as the meso-model of. Nevertheless, the present model predicts the breaking load with higher accuracy than the micromechanics based model of[START_REF] Zhang | Progressive Damage and Failure Modeling in Fiber-Reinforced Laminated Composites Containing a Hole[END_REF] which usually are more sophisticated computationally than the meso-scale models. It is important to note that the solution aborted prematurely in case of the cross-ply [(0/90) 6 ] s laminate due to the brittle fracture of the fiber in the on-axis plies. As a consequence, degree of non-linearity i.e [(±45) 6 ] s . In such cases plasticity or other non-linear constitutive law (for instance, a cubic polynomial was used in(Falzon, 2011a) ) is inevitable to incorporate in the numerical model in order to return accurate stress values to the in-situ failure theory. The softening of the built-in damage model is linear in contrast to the present model which deteriorates the material integration points exponentially as rendered in Figure2-6.

Figure 2 - 6 :

 26 Figure 2-6: Present model and ABAQUS built-in elastic/damage model for OHT [(±45) 6 ] sThe fiber and matrix damage initiation and accumulation of the laminate [(0/±45/90) 3 ] s is discussed briefly. The first cracking which took place in the laminate was the matrix cracking in the 90° ply. It was initiated near the hole corresponding to the reaction force of 12800 N (approximately 50% of the numerically predicted ultimate load) after the application of the incremental displacement load. The matrix cracking onset load is nearly the same as mentioned in[START_REF] Lee | Initial and progressive failure analyses for composite laminates using Puck failure criterion and damage-coupled finite element method[END_REF] due to the application of similar failure theory. Remind that this matrix cracking represents the meso-damage that was referred previously as point B in Figure1-16-a, IFFs of Puck's theory, which is the result of the coalescence of the number of diffused micro-cracks. Such micro-cracks are induced very early in the matrix by the exerting load at smaller magnitude as noted by point A on the load displacement curve in Figure 1-16-a. After obtaining saturation of IFFs in a single ply (saturation crack density), the local delamination commences as idealized in Figure 2-7.

Figure 2 - 7 :

 27 Figure 2-7: Transverse crack growth and local delamination after the saturation crack density in [0/90°] s laminate (Paris et al., 2010)

Figure 2 -

 2 Figure 2-8: Matrix damage accumulation in 90° ply at (a) 12800 N, (b) 20281 N (c) ultimate load, 25394 N

Figure 2 - 9 :

 29 Figure 2-9: Fiber damage progression in the 0° ply at (a) 19169 N, (b) 22263 N (c) ultimate load, 25394 N The numerical prediction of the present model with the results of the Lee et al. (Lee et al., 2015) are compared in Figure 2-10 and Table 2-7 with the experiments (Chang,

Figure 2 - 10 :

 210 Figure 2-10: Present model in comparison with Lee et al. (Lee et al., 2015)Table 2-7: Numerical results comparison of OHT made of T300/1034-C Layup Present (MPa)

Figure 3 -

 3 Figure 3-1: (a) Plane stress case, (b) transition from plane stress into 3D state of stress In such circumstances the requirement of 3D failure analysis comes into play because the out-of-plane strength of the composite structures is usually lower than the in-plane strengths. Therefore, it must be considered during the design and analysis phase. The impetus behind organizing the WWFE Part B (Hinton, 2013) is the 3D performance

  in which Puck's theory has been recommended based on the results in bi-axial and tri-axial loadings. It is based on Mohr-Coulomb fracture hypothesis which is appropriate for brittle fracture behavior of composite materials (e.g, concrete), and it is available both in 2D and 3D forms. It considers different failure behavior in combined normal and shear stresses of the material surfaces in the same manner as Mohr-Coulomb theory in the materials where the compressive strength exceeds the tensile strength. Puck et al. concluded based on the experiments that in case of transverse compression, the shear stresses have to cause fracture against an additional

  damage variables in the material principal axis i.e in 1-2-3

  Puck et al.(Puck, 2002a) has termed as 'stressing'. ( ) -thickness shear stress (t stands for the transverse direction shear stress). Due to the fact that the stressing on the fracture plane has three components, therefore, three strength parameters are required for the failure envelope to provoke the damage onset. These fracture strengths are denoted as: At

  out-of-plane shear strength of the action plane A R ⊥⊥ is, consequently, determined from the transverse compressive test. As at 45° in this test, the shear stress nψ τ is no doubt maximum being the maximum shear stress plane, but the normal compressive stress prevails too, to impede the fracture. Consequently, the fracture plane in such test is normally 53°±2° instead of 45° for continuous carbon/epoxy materials system.

Figure 3 - 3 :

 33 Figure 3-3: Fracture planes of the transverse cracks (IFF) corresponding to the type of stress (Matthias Deuschle, 2012)

IFFs

  σ = � . The residual strength of the fracture plane against IFFs can be determined as

Figure 3 - 4 :

 34 Figure 3-4: Weakening provoking range

Figure 3 -

 3 Figure 3-5: Shrinkage of IFF failure surface due to FF: Puck's Cigar

Figure 3 - 6 :

 36 Figure 3-6: Representation of the stress exposure factor as the ratio of the pseudo vector (Matthias Deuschle, 2012) Referring to Figure 3-7 and the state of stress there-in defined in the figure, based on the analytical calculation, point P has the maximum global exposure factor in transverse compression: , , E IFF f -, of 0.95 at angle of 75° which signifies that the ply is on the verge of the occurrence of IFF in compression. Now the same state of stress is applied on a cube of 1 mm side in ABAQUS using a single 8-node element C3D8 with the step size of 0.1° was used for the stress transformation. The contours plot of the action plane angle obtained in ABAQUS has shown in Figure 3-7. The fracture plane orientation through UMAT is in the range of 74.2°-76.3° represented by the state variable 8 (SDV8).

Figure 3 -

 3 Figure 3-7: Searching the orientation of the potential Puck's fracture plane In case of pure transverse compression ( 22 0 σ ≠ , and all other stress components are zero) the fracture plane determined experimentally by Puck et al. (Puck, 2002a) is usually oriented at 53°±2° in CFRPs. A compressive displacement load was applied transverse to the fiber direction. The fracture angle determined through the current model is in the range of 54.13°-58.97° denoted by state variable 8 (SDV8) as given in Figure 3-8 which is closed to the experimental fracture angle. The result can be improved further by using a finer angle step size in the iterative procedure. Similar fracture angle results have been found in (Reinoso et al., 2017).

Figure 3 - 8 :

 38 Figure 3-8: Fracture plane angle fp θ in pure transverse compression

4 (

 4 compression load step) is same to the slope of the line 0-2 that depicts the stiffness which has been degraded from the loading step path in tension shown by the points 0-1-2 in Figure3-9. When the load in compression exceeds the compressive strength, or in other words the stress exposure factor of Puck's theory in compression turns into unity, then further degradation takes place shown by the load path 0-4-5. This degradation is interpreted by the reduced slope of the line 0-5. As the cracks in tension open so, therefore, after reversing the load from point 5 to point 7, the residual stiffness is used in the analysis while applying the tension load step designated as 3-7. From point 7 onwards, the Puck's failure index in tension again becomes unity and further degradation in the materials stiffness in the fiber direction is accomplished by the model. Ultimately, the stiffness in fiber direction is completely vanished noted by the point 8 in

Figure 3 - 9 :

 39 Figure 3-9: Load reversal in fiber direction

- 10 .

 10 The transverse direction load reversals are illustrated by the load points 0-1-2-3-4-5-6-7-8 in Figure3-10 following the cracks closing framework. Material is degraded by the exerting load denoted by the load path 0-1-2 in tension. This residual stiffness is used in compression load step also based on the approach of no crack closing. The complete crack closing concept states that the undamaged stiffness shall be used as accomplished by the model illustrated by load curve 3-4. While going back from compression to tension, the cracks open again and hence the residual stiffness is used while computing the load displacement curve.

Figure 3 - 10 :

 310 Figure 3-10: Load reversal in transverse direction Moreover, different recovery factors are used which signify the weightage of the degradation in the load reversals due to the fact that the stiffness recovery is more pronounced when the load comes back from tension into compression rather than changing sign from compression into tension.

  simulations were run to generate the failure envelope for the IFFs initiation and are plotted in Figure3-11 where the failure stress exposure factor is shown for each laminate orientation.

Figure 3 -

 3 Figure 3-11: Intra-laminar damage initiation in transverse tension and compression (each arrow shows the stress exposure factor 1 E f = )

Figure 3 -

 3 Figure 3-12: Three principal sliding direction with respect to fiber, (a) Parallel (P), (b) Anti-Parallel (AP), (c) Normal (N) (Rodríguez-Temblequ, 2014)

Figure 3 -Figure 3 -

 33 Figure 3-13: (a) SEM-micrograph of a parallel oriented composite surface scratched by a diamond tip SEM, (b) an enlarged segment of the contact area in the case of P-fiber orientation (Friedrich et al., 2005) Figure 3-14 show the typical wear mechanisms in the case of AP-fiber orientation. Due

Figure 3 -

 3 Figure 3-15: Typical wear mechanisms of a parallel oriented unidirectional FRPs composite

Figure 3 -

 3 Figure 3-17: SEM (a) Sliding parallel to fiber (P), (b) sliding perpendicular to fibers (AP)[START_REF] Sharma | Influence of orientation of long fibers in carbon fiberpolyetherimide composites on mechanical and tribological properties[END_REF]

Figure 3 -

 3 Figure 3-18: Modeling sliding (a) sliding parallel to fiber, (b) sliding perpendicular to fibers In this study, we have considered composites based on Polyetherimide resin and UD reinforced carbon fibers (Polyacrylonitrile-PAN based high modulus) in 80% by volume that were developed by compression molding after optimization of various parameters (Sharma et al., 2010). Composites with different orientation of fibers were characterized for various properties such as tensile (ASTM D-638), in-plane shear strength (Iosipescu-ASTM D-5379/D 5379M), and inter-laminar shear strength (ILSS)

Figure 3 -

 3 Figure 3-19 explains the geometry, mesh constructions, and the Boundary Conditions (BCs) adopted in the model. In order to reduce the CPU time for this model, only the zone of interest is simulated in the specimen. Hence, the sample, that has dimension of 0.5 0.5 1 × × mm 3 , was divided into three rectangular parallelepiped parts.

Figure 3 -

 3 Figure 3-19: Simulation set-up (a) BCs, (b) FEM mesh

Figure 3 -

 3 Figure 3-20: Von Mises equivalent stress distribution for P and AP configurations , (a) indentation step, (b) sliding steps

  -21 and, only the stresses during the sliding step are given. It is pertinent to mention that for reasons of image quality, the mesh is not represented rather a view cut is used to visualize the stress in the depth of the sample. The dotted line indicates the momentary position of the sliding counterpart with respect to the composite sample.

Figure 3 -

 3 Figure 3-21: Normal stress distribution for P an AP sliding steps

Figure 3 -

 3 Figure 3-22: Shear stress contours in P and AP sliding

Figure 3 -Figure 3 -

 33 Figure 3-24. In case of anti-parallel configuration, these exposure factors are given in Figure 3-26 and Figure 3-27.

Figure 3 -

 3 Figure 3-23: Puck's FFs damage exposure factors: Parallel configuration

Figure 3 -

 3 Figure 3-24: Puck's IFFs damage exposure factors: Parallel configuration

Figure 3 -

 3 Figure 3-25: (a) A single wear groove on a polished P-oriented CF/PEEK-surface, as produced by a diamond indenter under a load of 3 N, (b) details of the failure mechanisms in the central part of the groove[START_REF] Friedrich | Finite element analysis of a polymer composite subjected to a sliding steel asperity[END_REF]

Figure 3 -

 3 Figure 3-26: Puck's FFs damage exposure factors: Anti-Parallel (AP) configuration In case of AP-configuration, the same fiber exposure factors distributions are observed for the indentation and sliding steps. No tensile failure of the fibers is obtained because

Figure 3 -

 3 Figure 3-27: Puck's IFFs damage exposure factors: Anti-Parallel (AP) configuration Concerning f E,IFF,-, the sliding leads to an increase of 69% in this particular exposure factor. Its distribution is much closed to the one obtained in the case of parallel configuration. f E,IFF,-factor can be regarded as a good indicator of the damage in matrix due to the sliding contact where both shear and transverse compression play their roles in the damage development. Experimentally, it is known that if AP-fiber orientation scratch test is considered, the shear and the tension/compression type de-bonding are the most dominant wear mechanisms (Figure 3-28).

Figure 3 -

 3 Figure 3-28: A single wear groove on a polished AP-oriented CF/PEEK-surface, as produced by a diamond indenter (Friedrich et al., 2002)If we consider various fiber orientation angle (0°, 15°, 30°, 45°, 60°, 75, 90°) with respect to the sliding direction, damage exposure factors can be evaluated with the

  Figure 3-30, the specific Puck damage indicator associated with f E,IFF,-as a function of the normal load and the fiber orientation angle is shown. It is evident that the same tendency as compared to the specific wear rate is obtained.

Figure 3 -

 3 Figure 3-30: Specific Puck damage indicator (f E,IFF,-/ F) as a function of fiber orientation angleIn particular, the specific damage obtained for the anti-parallel configuration (fiber orientation angle of 90°) is very high compared to the parallel configuration (fiber orientation angle of 0°). This can be explained by the analysis of the stresses and the Puck's criterion in each configuration. Figure3-31 is a simplified representation of the

Figure 3 -

 3 Figure 3-31: Simplified representation of the stresses on the contact surface bounded by the vertices A, B, C and D in the two sliding configurations It is easily observed that for the same applied contact pressure (σ 22 < 0), the strength in the case of parallel configuration is higher than in the anti-parallel one. In the graphs of the Figure 3-29 and Figure 3-30, wear rate and the specific Puck damage indicator

Figure 3 -

 3 Figure 3-32: Specific Hashin compressive matrix damage indicator as a function of fiber orientation angle

  visualized by comparing the failure envelopes of the Puck's theory which is based on the Mohr-Coulomb hypothesis with Hashin's theory. In Figure 3-33, for a contact pressure between 0 and 58 MPa (i.e. -58 MPa<σ 22 <0 MPa) the shear strengths 12 σ and 23 σ increase to 38 MPa and 28 MPa respectively.

Figure 3 -

 3 Figure 3-33: Puck and Hashin criteria (a) in 22 σ , 12 σ stress space, (b) 22 σ , 23 σ stress space

Figure 4 - 1 :

 41 Figure 4-1: Carbon fiber dry fabric and in-plane polyester (PES) stitchingThe desired target fiber volume fraction was chosen as 55%. The wet ply thickness calculated based on the desired fiber volume fraction is 0.156 mm. For the fabrication of 1 mm thick plate 6.39 plies are required. So total six plies were stacked to obtain a plate for the tension test in fiber direction. The fiber volume fraction was recalculated which was 52%. The fabrication of [0°] 6 laminate having nominal thickness of 1 mm for the fiber direction tension test is depicted in Figure4-2.

Figure 4 - 2 :

 42 Figure 4-2: (a) Resin-infusion fabrication process, (b) [0°] 6 cured plate after unmolding Following the manufacturer recommendations, all the plates fabricated were unmolded after 24 hours and were cured for 8 hours maintained at 60°C in a programmable curing oven as shown in Figure 4-3.

Figure 4 - 3 :

 43 Figure 4-3: 60°C curing for a duration of 8 hoursFurthermore, rectangular tabs of 2 mm thickness were cut from aluminum alloy panels, and then were bonded at the ends of all the specimens to transfer the load smoothly,

Figure 4 - 4 :

 44 Figure 4-4: ASTM 3039 specimens, (a) 0°, (b) 90°

Figure 4 -

 4 Figure 4-5: (a) [0°] 6 tensile CFRPs coupon with extensometer installed, (b) catastrophic fracture of one of the specimens, (c) post-test fractured specimensOne of the stress-strain curves in the fiber direction is plotted in Figure4-6 which demonstrates a linear behavior in the fiber direction. The material properties determined are described in Table4-2.

Figure 4 - 6 :

 46 Figure 4-6: Experimentally determined longitudinal stress-strain curve in tension In order to obtain 2 mm thick composite specimen for the 90° testing, 12 plies were embedded: [90°] 12 . The pre-test and post-test specimens are presented in Figure 4-7.

Figure 4 -

 4 Figure 4-7: (a) [90°] 12 tensile CFRPs coupons, (b) specimen with extensometer installed, (c) post-test fractured specimens

Figure 4 - 8 :

 48 Figure 4-8: Experimentally determined transverse tension behavior of one of the five specimens

Figure 4 - 9 :

 49 Figure 4-9: Layout of the specimen [±45°] 2s

  as zero, b and d are the width and thickness of the specimen respectively. It is emphasized that at least three strain gauges are required to define the state of strain in plane stress case but if the principal axes are known in the experiments then two strain gauges are sufficient. Besides the strain gauges, Digital Image Correlation (DIC) was used here to record strain field. The longitudinal stress and normal strain in x and y directions recorded with DIC are plotted in Figure4-10. In such

Figure 4 - 10 :Figure 4 -

 4104 Figure 4-10: Longitudinal stress xx σ and the strains in x and y directions of [±45°] 2s

Figure 4 - 4 . 3 . 3

 4433 Figure 4-12: (a) Fractured specimen No.4, (b) percentage normal strain field in load direction ( ) yy ε

Figure 4 -

 4 Figure 4-13: Loading-unloading curve 4.3.4 OHT tensile testing

Figure 4 -

 4 Figure 4-15 describes the OHT failed sample and the strains in x and y directions at rupture recorded by DIC. The matrix failure in [±45°] 4s is aligned with the fiber direction along the 45° angle. Cross-ply laminate failed completely at breaking stress of 453.0 MPa. This stress is also calculated based on the net section of the specimen at the center. Figure 4-16 depicts the fractured cross-ply specimens and the strain in the x and y directions.

Figure 4 -

 4 Figure 4-14: Longitudinal stress xx σ and the strains in x and y directions of [±45°] 4s OHT

Figure 4 -Figure 4 - 17 :

 4417 Figure 4-16: [0/90°] 4s OHT tension testing, (a) Fractured sample, (b)

Figure 4 -

 4 Figure 4-18: Bi-axial Arcan test fixtures, (a) shear coupled with tension, (b) shear coupled with compression (Van Den, 2007)

Figure 4 -

 4 Figure 4-20: Backside of the newly designed Arcan fixture, (a) Guide shaft locking position, (b) linear ball bearings used for guide shaft support

  . The objectives of the research work of El-Hajjar et al. were to use an optimum geometry of the Arcan specimen for two directions of UD laminate: axial and transverse directions with shear load application only. Arcan specimen of radius of 2.54 mm and angle 90° was proposed based on the optimization.

Figure 4 -

 4 Figure 4-23: (a) Layout of optimized Arcan specimen for UD laminate , (b) assessment of the normalized shear stress distribution, (c) Pure shear stress-strain using the gauge rosettes (El-Hajjar, 2004) Here in the present work, first the specimen in Figure 4-23 was simulated for the UD [0°] 16 called as transverse orientation in ABAQUS/Standard linear elastic constitutive law. The material properties are given in Table 4-5.

  profile having thickness of 2.5 mm was modified to be compatible in the assembling with the new Arcan fixture proposed here (Figure4-21-b). The boundary conditions (BCs), FEM discretization, and the shear stress distribution are given in Figure4-24 for the UD Arcan specimen having modified geometry but similar notch properties of El-Hajjar et al.

Figure 4 -

 4 Figure 4-24: (a) Load and BCs, (b) FEM meshing, (c) [0°] 16 UD Arcan (El-Hajjar et al. notch parameters), (d) optimized Arcan for cross ply laminate [0/90°] 4s Similar study was conducted here for the cross-ply laminate [0/90°] 4s to aim the uniform shear stress distribution by varying the notch root radius and angle. For this purpose, the ABAQUS scripting capability with Python was used in plane stress case which is best suited for parametric optimization analysis. The optimized Arcan specimen for the cross ply [0/90°] s is given in Figure 4-25-a where the notch radius is 2.38 mm and the angle is 102.7°. The pure shear loading configuration is also idealized in Figure 4-25-b whereθ is 90° with respect to the x-axis. It is important to describe that the holes in this modified Arcan specimen are far away from the notch zone as compared to the Arcan specimens used in (El-Hajjar, 2004) in order to avoid fracture in the holes.

Figure 4 -

 4  describes the shear stress field for this optimized geometry. The normalized shear stress, as referred in Figure4-26, was computed by dividing the shear stress at the significance section of the Arcan specimen with the applied shear stress. It can be noticed that a considerable segment of the significant section is in uniform shear.

Figure 4 -

 4 Figure 4-25: (a) Optimized Arcan specimen for [0/90°] 2s , (b) Pure shear test set-up

Figure 4 -Figure 4 -

 44 Figure4-27: Shear stress-strain response from pure shear test[START_REF] El-Hajjar | In-plane shear testing of thick-section pultruded FRP composites using a modified Arcan fixture[END_REF] These Arcan specimens will be used for the pure shear stress-strain behavior and will be compared with the shear stress-strain curve obtained in tension test of [±45°] 2s in addition to bi-axial loading. All the plates for Arcan testing were manufactured by handlayup method and same curing conditions were executed as for the ASTM characterization plates in section 4.3. The butterfly specimens were cut with precise tolerance water jet cutting machine from the plates. Five Arcan samples with strain gauges installed for two configurations of [0°] 16 and [0/90°] 4s each have been shown in Figure4-28. In the top most and bottom plies in the cross-ply laminate [0/90°] 4s , fibers are parallel to the notch-to-notch direction.

Figure 4 -

 4 Figure 4-28: Two Arcan configurations, [0°] 16 and [0/90°] 4s

Figure 4 -

 4 Figure 4-29: Experimental set-up

Figure 4 -

 4 30. xx σ denotes the external uniaxial stress. Longitudinal stress 11 σ decreases from its maximum value at θ = 0°and ultimately becomes zero at θ = 90 ± °. The transverse normal stress 22 σ variation is opposite to the variation in 11 σ . Shear stress 12 τ gains its maximum value at 45 ± °.

Figure 4 - 4 . 6 . 1

 4461 Figure 4-30: Stress components induced internally as a function of the fiber angle when a uniaxial external stress xx σ is applied

Figure 4 -

 4 Figure 4-31: Specimen in pure shear stress Two strain gauges of 6 mm gauge length and 10 mm total length including the soldering legs were glued along the x-axis and y-axis with tolerance in resistance of ±0.50 Ω and maximum resistance capacity of 118.6 Ω supplied by Tokyo Measuring Instrument Lab (TML Co.). Gauge factor of 2.13 was used for the conversion of relative resistivity into strain. The zone of uniform shear in the optimized Arcan specimen in this study is 10 mm shown in Figure4-25. Therefore, the size of the pure shear zone is adequate for the strain gauges to be mounted.

Figure 4 -

 4 Figure 4-32 illustrates the shear stress-strain curves obtained from three different tests (i) tension test of [±45°] 2s complaint with ASTM D3518-21, (ii) [0°] 16 in pure shear

Figure 4 -

 4 Figure 4-32: Comparison of shear stress-strain curves based on two different test methods: Arcan and tension test of [±45°] 2s

Figure 4 -

 4 34 (b) describes the shear strain distribution at final failure which is -5.4%.

Figure 4 - 4 . 6 . 2

 4462 Figure 4-33: Post-test Arcan fractured specimen of cross-ply in pure shear, (a) Damage zone at the notch section , (b) Macroscopic delamination, (c) Specimen failed by pure shear

Figure 4 -

 4 Figure 4-35: Bi-axial test with Arcan, (a) test set-up of [0°] 16 at 45° to the load, (b) Failed specimen, (c) Bi-axial stress comparison with pure shear of [0°] 16 and [0/90°] 4s laminates Similarly, [0/90°] 4s specimens were tested in bi-axial stress case. Failure was observed at significant section as shown in Figure 4-36 (a), while Figure 4-36 (b) compares the shear stress-strain curve in pure shear case and in combined stress case. The failure shear stress obtained in the bi-axial experiment was 37.31 MPa at shear strain of 1.56%which is inferior to the pure shear test of a similar specimen.

Figure 4 -

 4 Figure 4-36: (a) Fractured specimen of [0/90°] 4s , (b) Shear stress-strain plots of pure shear and combined shear/normal stress

Figure 4 -

 4 Figure 4-37: Cross-ply pure shear simulation set-up (a) meshing and BCs, (b) stacking sequence

Figure 4

 4 Figure 4-40.

Figure 4 -

 4 Figure 4-38: Model prediction for [0/90°] 4s in case of pure shear

Figure 4 -Figure 4 -

 44 Figure 4-40: Mode-A damage initiation and saturation, shear damage variable 21 ( ) d

Figure 4 -

 4 Figure 4-42: On-axis tension test with Arcan fixture of [0°] 16 laminate, (a) Fractured specimen showing splitting type failure mechanism, (b) Longitudinal stress-strain curve In tension test of cross-ply laminate, the final failure of the specimen occurred due to fractures at the bolted holes as depicted in Figure 4-43. But, partial matrix failure was also observed in the notch section. The stresses at these failed clamped holes exceeded the strength of the bolted joints. The stress-strain curve shows non-linearity which may be due to the matrix failure in the 90° ply.

Figure 4 -

 4 Figure 4-43: On-axis tension test with Arcan fixture of [0/90°] 4s laminate (a) post-test specimen where partial failure was observed at the significant section, (b) longitudinal stress-strain curve

Figure 4 -

 4 Figure 4-44: Failure stresses for two laminates [0°] 16 and [0/90°] 4s Damage coupling in cross-ply 4.7

Figure 4 -

 4 Figure 4-45: [0/90°] 4s specimens in pure shear stress stateThe tensile stress-strain curves of the two specimens along with the specimen having no initial damage are compared in Figure4-46. It can be noticed that the damage in shear has influenced the response of the Arcan specimen in tension. The dominant failure mechanisms are fiber/matrix de-bonding and matrix cracking when FRPs are loaded in shear. These failures degrade the plies transverse and shear elastic properties. Here in this case, the difference of the tensile mechanical behavior is caused by these two failure mechanisms. The fiber direction elastic constants are less affected by the failure in shear load but as this laminate is cross-ply, therefore, the 90° ply has contributed less in carrying the load. Hence, the 90° plies which are damaged caused this difference. In contrary, the ply which is aligned to the load in the second step of tensile load in Arcan fixture is assumed to carry the same load because the failure mechanisms of the first shear step are not very influential on the fiber response.

Figure 4 -

 4 Figure 4-46: [0/90°] 4s laminate failure in tension

  at rupture ..............................................................................................................................Figure 4-17: OHT tests of [0/90°] 4s and [±45°] 4s laminates ..................................................................... Figure 4-18: Bi-axial Arcan test fixtures, (a) shear coupled with tension, (b) shear coupled with compression (Van Den, 2007) .................................................................................................................

	yy ε =	3.8%

  ...................................................................................... Figure 4-32: Comparison of shear stress-strain curves based on two different test methods: Arcan and tension test of [±45°] 2s ............................................................................................................................. Failure shear strain -5.4% ....................................................................................................... ................................................................................................................................................. Fractured specimen and damage contour (Mode-A, Puck's theory) of cross-ply in pure shear ................................................................................................................................................................. Figure 4-42: On-axis tension test with Arcan fixture of [0°] 16 laminate, (a) Fractured specimen showing splitting type failure mechanism, (b) Longitudinal stress-strain curve .................................................... Figure 4-43: On-axis tension test with Arcan fixture of [0/90°] 4s laminate (a) post-test specimen where partial failure was observed at the significant section, (b) longitudinal stress-strain curve .....................

	Figure 4-41: Figure 4-44: Failure stresses for two laminates [0°] 16 and [0/90°] 4s ........................................................
	Figure 4-45: [0/90°] 4s specimens in pure shear stress state ......................................................................
	Figure 4-46: [0/90°] 4s laminate failure in tension ....................................................................................
	Figure 4-33: Post-test Arcan fractured specimen of cross-ply in pure shear, (a) Damage zone at the notch
	section , (b) Macroscopic delamination, (c) Specimen failed by pure shear ............................................
	Figure 4-34: Shear strain field acquired by DIC (a) Uniform shear distribution in notch area at shear strain
	-2.4%, (b) Figure 4-35: Bi-axial test with Arcan, (a) test set-up of [0°] 16 at 45° to the load, (b) Failed specimen, (c)
	Bi-axial stress comparison with pure shear of [0°] 16 and [0/90°] 4s laminates .........................................
	Figure 4-36: (a) Fractured specimen of [0/90°] 4s , (b) Shear stress-strain plots of pure shear and combined
	shear/normal stress ...................................................................................................................................
	Figure 4-37 Cross-ply pure shear simulation set-up (a) Meshing and BCs, (b) Stacking sequence ........
	Figure 4-38: Model prediction for [0/90°] 4s in case of pure shear ...........................................................
	Figure 4-39: Shear strain contour at failure (a) the top ply of Arcan specimen recorded by DIC, (b) FEM
	shear strain field at the top ply (fiber along y-direction) (c) shear strain field in ply 2 (fiber along x-
	direction ..Figure 4-40: Mode-A damage initiation and saturation, shear damage variable 21 ( ) d ...........................
	xxi xxii

) .................................... Figure 4-28: Two Arcan configurations, [0°] 16 and [0/90°] 4s .................................................................. Figure 4-29: Experimental set-up ............................................................................................................ Figure 4-30: Stress components induced internally as a function of the fiber angle when a uniaxial external stress xx σ is applied .....................................................................................................................

Figure 4-31: Specimen in pure shear stress .......

Table 2 -

 2 1: 2D Puck's failure mode invoking conditions ............................................................................ Table 2-2: AS4/PEEK properties ............................................................................................................... Table 2-3:Puck's criterion properties for CFRPs ....................................................................................... Inclination parameters .............................................................................................................. Table 3-2: AS4/3501-6 .............................................................................................................................. Table 3-3: Engineering constants and mechanical strengths (Sharma et al., 2010) ................................... Table 3-4: Contact parameters during the indentation phase ..................................................................... Table 3-5: Damage exposure factor with respect to the fiber angle ...........................................................Table 4-1: CFRP constituent's material specifications from suppliers ...................................................... Table 4-2: Tensile direction material properties ........................................................................................ Table 4-3: Transverse direction material properties .................................................................................. Table 4-4: In-plane shear properties .......................................................................................................... Table 4-5: Elastic properties E-glass (El-Hajjar, 2004) ........................................................................... Table 4-6: Elastic constants and model input parameters ........................................................................

	Table 2-4: T300/1034-C properties............................................................................................................
	Table 2-5: OHT breaking stress comparison of T300/1034-C ...................................................................
	Table 2-6: OHT breaking stress comparison of T300/1034-C ...................................................................
	Table 2-7: Numerical results comparison of OHT made of T300/1034-C ................................................
	Table 3-1:

Table 2 -

 2

1: 2D Puck's failure mode invoking conditions E f invoking condition Mode of failure

Table 2 -

 2

	2: AS4/PEEK properties
	0 11 E	E	0 22	0 21 G (MPa)	12 ν	X	T	X	C	T Y	C Y	S	21
	(MPa)	(MPa)					(MPa)	(MPa)	(MPa)	(MPa)	(MPa)
	127600	10300	6000	0.32 2023.0	1234.0	92.7	176.0	82.6
	22 T 	11 T 	21 	ν	f	21	a		β		α	η	11f E
	(N/mm)	(N/mm)	(N/mm)										(MPa)
	5.6	128.0	4.93	0.20	2.0	295.0274 0.142857 0.0002 225000

Table 2 - 3

 23

:Puck's criterion properties for CFRPs

Table 2 -

 2

	4: T300/1034-C properties
	0 11 E	E	0 22	0 21 G	12 ν	X	T	X	C	T Y	C Y	S	21
	(MPa)	(MPa)	(MPa)			(MPa)	(MPa)	(MPa)	(MPa)	(MPa)
	146800	11400	6100	0.30	1730	1379	66.5	268.2	58.7
	22 T 	11 T 	21 	ν	f 21	a		β		α	η	11f E
	(N/mm)	(N/mm)	(N/mm)									(MPa)
	0.23	89.83	0.46	0.20	1.25	567.9092 0.272405 0.0002 230000

Table 2

 2

	-5: OHT breaking stress comparison of T300/1034-C
	Layup	Present	Chen et al.	Vasiukov et al.	Experiment	Error
		(MPa)	(MPa)	(MPa)	(Tan SC, 1991)	(%)
					(MPa)
	[0/(±45) 3 /90 3 ] s	232.53	238.30	231.47	235.80	-1.38
	[0/(±45) 2 /90 5 ] s	189.10	205.83	190.59	185.47	1.95
	[0/(±45) 1 /90 7 ] s	150.40	135.67	-	159.96	-5.97

  In contrast here, four stress exposure factors exist as referred in eq.(3.2). It shall be noted that the matrix related expressions are the functions of the angleθ which denotes the orientation of the Puck's fracture plane with respect to the out-of-plane normal in the laminate thickness direction.

	, ,	t	11	(	22	33	)		11
	, ,	c	11		(	22	33	)	11
				t
	,	,	At	At

The Puck's theory stress exposure factors are based on the effective stress components in the fracture plane coordinate system. As discussed in the last chapter that 2D formulation of this theory has five failure indices or stress exposure factors.

Table 3 -

 3 1: Inclination parameters

	Material	t p ⊥⊥	c p ⊥⊥	t p ⊥ �	c p ⊥ �	m σ	f
	GFRPs	0.30	0.25	0.20	0.25	1.3
	CFRPs	0.35	0.30	0.25	0.30	1.1

Table 3 -

 3

	2: AS4/3501-6
	0 11 E	0 22	0 33
	(MPa)

Table 3 -

 3

	4: Contact parameters during the indentation phase
	Case	δ (µm)	2a (µm)	2b (µm)	P max (MPa)
	P-configuration	2.249	60	70	638
	AP-configuration	2.274	70	60	685

Table 3 -

 3 5: Damage exposure factor with respect to the fiber angle Based on these observations, it is possible to find a relationship between the damage exposure factors and the wear? To study the wear and the tribo-performance tests, different experimental set-ups are used. Pin-on-disc is one of the commonly followed tests for this purpose. It consists of a plane square stationary pin made of the sample material which is fixed and a normal force is applied. A disc is rotating and is in continuous contact with the sample pin. The coefficient of friction mis determined by the division of tangential force by normal applied force: Normal force is known in the experiment while the tangential force is recorded with load sensor.Sharma et al.

	Fiber orientation	f E,FF,+	f E,FF,-	f E,IFF,+	f E,IFF,-
	0°	1.021	1.732	7.033	4.279
	15°	1.010	1.719	9.779	4.79
	30°	0.945	1.684	12.98	5.099
	45°	0.832	1.640	17.21	6.011
	60°	0.681	1.597	21.73	6.396
	75°	0.535	1.572	24	6.188
	90°	0.406	1.567	24.32	5.677
	Puck's criterion seems to well predict the contact damage occurring during a scratch
	test.

Table 4 -

 4 1: CFRP constituent's material specifications from suppliers

	Manufacturer's Fabric reference number	30000914
	Fiber manufacturer		SAERTEX GmbH.
	Fiber type		T700SC 50C
	Carbon fiber density (g/cm	3 )	1.79
	GSM		154±5.0
	Filament Diameter		7.0 microns
	Polymer type and supplier:	Sicomin
	Thermoset polymer		Epoxy: SR 8100
			Hardener: SD 8824
		Laminate

Theoretical desired fiber volume fraction 55% Dry thickness per ply (mm) 0.086 Wet thickness per ply (mm)

	0.156

Table 4 -

 4 2.

Table 4 - 2

 42

: Tensile direction material properties 11 E (GPa)

Table 4 - 3

 43

: Transverse direction material properties 22 E (GPa)

Table 4 - 4

 44

	: In-plane shear properties
		12 G (GPa)	12 S (MPa) a	β	α
	Mean value	4.05±0.8	45.67±2.1 2.5 295.0274 0.2

Table 4

 4

	-5: Elastic properties E-glass (El-Hajjar, 2004)
	0 11 E	E	0 22	=	0 33 E
	(MPa)		(MPa)

Table 4 -

 4 6: Elastic constants and model input parameters

	0 11 E	E	0 22	0 21 G (MPa)	12 ν	X	T			S	21
	(MPa)	(MPa)					(MPa)			(MPa)	(MPa)
	115000	9800	4050	0.31 1950.0	1234.0	43.2	80.0	45.67
	22 T 	11 T 	21 	ν	f	21	a		β	α	η	11f E
	(N/mm)	(N/mm)	(N/mm)								(MPa)
	5.6	128.0	4.93	0.20	2.5	295.0274	0.2	0.0002 225000

C X (MPa) T Y (MPa) C Y

Acknowledgments

Chapter 2

Meso-scale progressive damage modeling the model underestimated the breaking strength with relatively higher error as referred in The experimental values in Table 2-6 have been reported in [START_REF] Chang | A Progressive Damage Model for Laminated Composites Containing Stress Concentrations[END_REF]. If the present model is used in the explicit scheme VUMAT of ABAQUS, then it is believed to produce more accurate results for this on-axis dominant plies laminate of cross-ply which was also followed in [START_REF] Zhang | Progressive Damage and Failure Modeling in Fiber-Reinforced Laminated Composites Containing a Hole[END_REF] to avoid the convergence problem in UMAT. The last case study has been selected from [START_REF] Lee | Initial and progressive failure analyses for composite laminates using Puck failure criterion and damage-coupled finite element method[END_REF] due to the reason that 2D

Puck's theory has been used with sudden degradation and elastic constitutive law.

Therefore, it is envisaged that significant improvement can be achieved if gradual degradation at the material integration points is used in combination with a more realistic constitutive law. Pronounced underestimation can be noticed in the results presented in [START_REF] Lee | Initial and progressive failure analyses for composite laminates using Puck failure criterion and damage-coupled finite element method[END_REF] due to sudden degradation except the cross-ply laminate.

ABAQUS/Standard has also a built-in elastic/damage model based on Hashin's failure theory [START_REF] Hashin | Failure criteria for unidirectional fiber composites[END_REF] which is available only for plane stress elements developed by [START_REF] Lapczyk | Progressive damage modeling in fiber-reinforced materials[END_REF]. The stress strain curve of the present model and the ABAQUS built-in model have been plotted in Figure 2-6 for laminate that poses higher

Damage exposure factor vs specific wear rate

Experimentally, adhesive wear study is conducted using pin-on-disc experimental setup, though there are other techniques also exist. For FRPs composites, pin is made of the composites of interest which is slid against a smooth disc of mild steel. After experiment, the pin is cleaned, dried and weighed to measure the loss in material by wear. The specific wear rate denoted by K 0 is calculated using the following equation.

Where ∆m is the weight loss in kg, ρ is the density in kg/m 3 , F is the applied normal load in Newton (N), and d is the distance slid in meters. Generally, this is performed for different load and velocity. Sharma et al. [START_REF] Sharma | Influence of orientation of long fibers in carbon fiberpolyetherimide composites on mechanical and tribological properties[END_REF] have studied the tribological performances of the considered material here which composed of unidirectional carbon fibers and thermoplastic Polyetherimide matrix (PEI), as a function of fiber orientation angle (0•, 15•, 30•, 45•, 60•, 75• and 90•) with respect to the sliding direction. The pin size was 10 x 10 x 5 mm 3 and three normal loads were applied viz. 200 N, 300 N and 400 N. For each test, the friction coefficient mwas recorded and is used here for the simulation.

In the experiments, they have observed that specific wear rate increased strictly with the fiber orientation as rendered in Figure 3-29. It was reported by the authors that fibers in transverse direction proved poorest in tribological characteristics. Moreover, very low wear rate was observed for composite with parallel fibers to the sliding direction. It is important to highlight that the main characteristics of this material is that in severe loading conditions (normal load of 400 N), wear decreased further confirming suitability of this composite as a dry bearing material for severe operating conditions (see .

A uniform pressure (corresponding to the normal force divided by the surface of the sample) and a tangential pressure (corresponding to the pressure multiplied by the friction coefficient) are applied uniformly to the contact surface. The assumption of uniform pressure is viable due to the fact that prior to the experiment, the pin was slid against a rough mild steel disc of R a value 0.3-0.4 µm to obtain a uniform contact. To compare the specific wear rate with the damage exposure factors qualitatively, we have Chapter 4 Two stainless steel guide shafts were also included that are supported by two linear ball bearings. The two linear ball bearings were constrained in the holes drilled in the upper half which is movable with the load cell. The two shafts were fixed with two grub screws at one end each with the lower half. All these arrangements are given in Figure 4-20. The upper half of the fixture traverses on these guide shafts. The linear ball bearing limit the bending and unwanted moments which could hamper the in-plane loading results in addition to reduce the friction and make the in-plane movements smooth and frictionless.

Mechanical response under quasi-static in-plane biaxial loading

The modified Arcan fixtures of Yen at al. [START_REF] Yen | Evaluation of a modified Arcan fixture for the in-plane shear test of materials[END_REF], and the one proposed in the present study are shown in Figure 4-21 whereas, Figure 4-22 , describes final integration with the tensile machine.

Chapter 5

Conclusions and recommendations

Numerical modeling

In the present thesis, numerical modeling was carried out based on the coupled plasticity and damage theory for FRPs composites. Based on the extensive comparisons with the experiments, the model is more efficient and accurate. The main building blocks of the models are the physically based failure theory, Puck's theory, and the coupled non-linear constitutive law. Puck's theory was chosen due to the fact that it not only predicts the damage accurately but also gives the orientation of the fracture plane which is considered as one of the important characteristics.

The model is fully capable to capture the non-linearity due to the irreversible strain in the matrix and thus returns the accurate stress state to the Puck's theory failure exposure factors. As here Puck's theory has been employed to trigger the intra-laminar damage and the subsequent damage evolution which only holds good for continuous UD FRPs based composite structures. Therefore, this model cannot be used for the damage onset and evolution in the textile plies keeping in view the limitation of Puck's theory.

However, it can be used for the damage study in a fiber yarn in a textile based laminate.

The damage accumulation patterns in the UD plies are well in agreement with the published articles. Moreover, due to the plane stress assumption, the model is only capable to predict the damage which is induced by the in-plane loading where no dominant out-of-plane stresses are acting. Consequently, such limitation will under predict the damage magnitude due to ignoring the damage contribution of the out-ofplane stress components particularly in cases where the out-of-plane stress is significant and cannot be ignored. So in the dominant out-of-plane stresses scenarios, this model shall not be used because it will over predict the structures performance by ignoring the role of the out-of-plane stresses that lead to material deterioration.

The model was enhanced to 3D by employing the 3D form of Puck's theory in order to obtain a better damage evolution due to the contribution of the out-of-plane stress