Rozprawy doktorskie na temat „Micropolar Cohesive Damage Model”

A multiple scale theory is developed for the prediction of damage evolution in heterogeneous viscoelastic media. Asymptotic expansions of the field variables are used to derive a global scale viscoelastic constitutive equation that includes the effects of local scale damage. Damage, in the form discrete cracks, is allowed to grow according to a micromechanically-based viscoelastic traction-displacement law. Finite element formulations have been developed for both the global and local scale problems. These formulations have been implemented into a two-scale computational model Numerical results are given for several example problems in order to demonstrate the effectiveness of the technique.

This thesis presents an extended cohesive damage model (ECDM) for simulating crack propagation in fibre reinforced composites. By embedding the cohesive zone model (CZM) into the eXtended Finite Element Method (XFEM)and eliminating the enriched degree of freedoms (DoFs), the ECDM defines the cohesive crack path in an implicit way in equilibrium equations and enables the local enrichments of approximation spaces without additional DoFs. The contribution from additional DoFs can be accounted via the DoFs elimination,which allows discontinuities to exist within a finite element rather than the element boundaries. To account for the evolution of cohesion before crack propagation, in this developed ECDM, a new equivalent damage variable with respect to strain field is introduced to avoid the appearance of enriched DoFs,and to substitute the conventional characterization in the approximation of displacement jump. This variable is achieved based on the energy dissipation during post-failure process to characterize the damage evolution. Therefore,the constant dissipation of fracture energy during failure process is guaranteed. Eliminating the enrichment by adopting a condensation technique, the ECDM is expected to provide significant superiority in computational efficiency when modelling crack propagation in materials. The performance of the present ECDM is demonstrated by the initial applications in simulation of crack propagation in homogeneous and heterogonous structures, which show that the developed ECDM works well when comparing to experiment work and XFEM analysis. Regarding the computational cost, the ECDM can ease the computational burden by more than 60% reduction in terms of CPU time without sacrificing numerical accuracy and robustness. The feasibility of the ECDM in capturing delamination migration within fibre reinforced laminated composites is verified. Good agreements with experimental work are obtained and the present model’s advantage in accuracy and numerical efficiency comparing to CZM based model is demonstrated. This work makes contribution to academic knowledge and technology translation by the following points: 1. It is the first time to theoretically derive the fully condensed equilibrium equations of the ECDM based on the framework of XFEM; 2. An equivalent damage variable with respect to strain field is introduced for characterizing the effects from enriched DoFs and cohesive traction, which avoids physical displacement jump in presenting strong discontinuities; 3. A significant improvement of computing efficiency in non-linear fracture analysis is achieved through eliminating the enriched DoFs required by XFEM; 4. The developed ECDM provides a highly efficient tool for academics and engineers in predicting detailed multicrack failure mechanism in engineering materials and structures; 5. The ECDM is developed using common computer language FORTRAN, which can be easily integrated into other FEM commercial packages.

The effect of the element length is examined in modelling crack growth in fatigue loading of an adhesive joint. This is done for a cohesive element using an expression for the damage evolution developed at the University of Skövde which is implemented using the UMAT subroutine in the FE-solver Abaqus. These analyses are done for pure mode I loading by analysing a DCB-specimen loaded by a pure moment. An expression is developed in which the critical element length is dependent on the geometry of the specimen (in the form of the wave number of the adhesive joint), the element length, the material properties of the adhesive (in form of the material parameters , , ), the load applied (in form of the stress in the crack tip), the time step used in the analysis and the crack growth rate. It is shown that the results converge by decreasing the element length and the time step used. Therefore an expression for the crack growth rate as a function of the remaining parameters can be determined. Another expression is thereafter developed for the element length needed in order to get a crack growth rate within a certain range of the critical element length. The results show a regular pattern but are not monotone. Therefor two different definitions of the critical element length are tested, either by defining the critical element length as the point where the error is greater than an arbitrary boundary of 1 % of a converged result or where a least square approximation of the error is within 1 % of the converged results. The first method shows a highly irregular result which makes it difficult to develop an expression out of these results. The second method on the other hand gives results that are predictable enough to develop a function out of them. This is done using a regression analysis with all parameters of a third order expression in order to get an expression.

Les travaux présentés dans ce mémoire s'inscrivent dans l'étude et l'amélioration des modèles d'endommagement continus régularisés (non locaux), l'objectif étant d'étudier la transition entre un champ d'endommagement continu défini sur l'ensemble d'une structure et un modèle discontinu de fissuration macroscopique.La première étape consiste en l'étude semi-analytique d'un problème unidimensionnel (barre en traction) visant à identifier une famille de lois d'interface permettant de basculer d'une solution non homogène obtenue avec un modèle continu à gradient d'endommagement vers un modèle discontinu de fissuration cohésive. Ce passage continu / discontinu est construit de telle sorte que l'équivalence énergétique entre les deux modèles soit assurée, et reste exacte quelque soit le niveau de dégradation atteint par le matériau au moment où cette transition est déclenchée.Cette stratégie est ensuite étendue au cadre 2D (et 3D) éléments finis dans le cas de la propagation de fissures rectilignes (et planes) en mode I. Une approche explicite basée sur un critère de dépassement d'une valeur « critique » de l'endommagement est proposée afin de coupler les modèles continus et discontinus au sein d'un même calcul quasi-statique par éléments finis. Enfin, plusieurs résultats de simulations menées avec cette approche couplée sont présentés
The present work deals with the study and the improvement of regularized (non local) damage models. It aims to study the transition from a continuous damage field distributed on a structure to a discontinuous macroscopic failure model.First, an analytical one-dimensional study is carried out (on a bar submitted to tensile loading) in order to identify a set of interface laws that enable to switch from an inhomogeneous solution obtained with a continuous gradient damage model to a cohesive zone model. This continuous / discontinuous transition is constructed so that the energetic equivalence between both models remains ensured whatever the damage level reached when switching.This strategy is then extended to the bi-dimensional (and tri-dimensional) case of rectilinear (and plane) crack propagation under mode I loading conditions, in a finite element framework. An explicit approach based on a critical damage criterion that allows coupling both continuous and discontinuous approaches is then proposed. Finally, results of several simulations led with this coupled approach are presented

The use of adhesive joints in industrial structural applications is currently growing. Nevertheless, the designing process is still a challenge for engineers because there is a lack of effective models to predict their behaviour. In particular, the use of flexible adhesives has been become a trend in the industry because of the advantages that they offer, and many research are ongoing in order to develop effective analytical models for its design. Current models, which are developed for compressible rigid adhesives, are unable to accurately predict the behaviour of flexible adhesives joints with low elastic modulus, incompressibility and large strain to failure. The more advanced analytical models take in account the elasticity of adhesive as a constant distribution and still they do not offer enough precision because they do not consider the influence of the stress state to the stiffness, the influence of the damage and the large deformations present in the adhesive layer.
Les unions adhesives són especialment interessants en aplicacions estructurals i el seu ús ha augmentat notablement en la industria moderna. No obstant, el disseny d’una unió adhesiva segueix essent actualment un repte per als enginyers, doncs no es disposa de models eficaços per a predir el seu comportament. Particularment, la utilització d’adhesius flexibles s’està estenent cada vegada méss degut als avantatges que aquests ofereixen, i actualment s’estan duent a terme nombrosos esforços per obtenir models analítics méss eficaços que permetin poder analitzar-ne el seu comportament. Els models existents, desenvolupats per adhesius compressibles i rígids no son capaços de predir amb precisió el comportament d’unions amb adhesius flexibles, que presenten un mòdul elàstic baix, incompressibilitat i grans deformacions fins a la seva ruptura. Els models analítics més avançats incorporen l’efecte de l’elasticitat de l’adhesiu en la seva formulació com una distribució constant, i com que no tenen en compte aspectes com l’efecte de les tensions sobre la rigidesa de l’adhesiu, la influència del dany i les grans deformacions a la capa de l’adhesiu no són suficientment precisos en la modelització del comportament de la unió.
Las uniones adhesivas son de especial interés en aplicaciones estructurales y su uso ha aumentado notablemente en la industria moderna. Sin embargo, el diseño de uniones adhesivas sigue siendo a día de hoy un reto para los ingenieros que carecen de modelos eficaces para la predicción de su comportamiento. En particular, el uso de adhesivos flexibles se ha convertido en una tendencia en la industria debido a las ventajas que estos ofrecen y se están dedicando numerosos esfuerzos para lograr modelos analíticos más eficaces. Los modelos existentes, desarrollados para adhesivos compresibles rígidos, no son capaces de predecir con precisión el comportamiento de uniones con adhesivos flexibles que presentan bajo módulo elástico, incompresibilidad y grandes deformaciones hasta la rotura. Los modelos analíticos más avanzados incorporan el efecto de la elasticidad del adhesivo en su formulación como una distribución constante y aún carecen de suficiente precisión debido a que no toman en cuenta efectos como la influencia del estado tensional sobre la rigidez, la influencia del daño y las grandes deformaciones presentes en la capa de adhesivo.
Lotura itsasgarriak interes handiko loturak dira aplikazio estrukturaletarako eta bereerabilera nabarmenki handitzen joan da industria mailan. Hala ere, gaur egun loturaitsasgarrien diseinuak erronka izaten jarraitzen du, ingeniariek horien portaera ezagutzekoeredu egokirik ez daukatelako. Konkretuki, itsasgarri malguen erabilera joera bilakatuda industria mailan, eskaintzen dituzten abantailengatik, eta hori dela eta ahalegin handiakegiten ari dira eredu analitiko eraginkorragoak lortzeko. Erabilgarri dauden ereduak,itsasgarri zurrun konprimigarrietarako garatuak zainik, ez dira gai modulu elastiko baxua,konprimitzeko ezintasuna eta haustura arteko deformazio handiak jasaten dituztenitsasgarri malguen jarrera iragartzeko. Eredu analitiko garatuenek itsasgarriaren malgutasunarenefektua banaketa konstante bezala moduan agertzen dute, eta oraindik zehaztasunfalta dute, zurruntasunaren tentsio egoeraren efektuak, kaltearen eragina eta itsasgarriangertatzen diren deformazio handiak kontuan hartzen ez dituztelako.

Le travail présenté a pour objectif la prédiction numérique de la résistance résiduellede grandes structures vis-à-vis d’évènements accidentels, tels que ceux rencontrés p.ex. dans le cas de la collision de navires ou d’impact d’oiseaux en aéronautique. Cesévènements peuvent dans certain cas conduire à la rupture, qui est ici considéréeductile. La difficulté de cette étude, consiste à reproduire dans une méthodologieunifiée basée sur la méthode des éléments finis les étapes successives menant àla ruine ultime de la structure. Ces étapes sont : l’endommagement ductile, lalocalisation de la déformation et la propagation de la fissure. Un élément essentiel pour la conception d’un modèle de fissuration ductile prédictif est le traitement numérique de la phase transitoire critique de localisation de la déformation induite par l’endommagement dans une bande de matière étroite.A cet effet, trois points de vue différents en termes de champ de déplacement àtravers la bande de localisation sont proposés. Ces trois approches se distinguentpar le type de discontinuité considérée : forte, faible et régularisée (expression nonlinéaire). Un cadre variationnel consistant est élaboré pour chacune des trois approches.Ainsi la cinématique enrichie est incorporée dans la formulation de l’élément fini enutilisant la méthode des éléments finis enrichis (X-FEM). Puis, la performance deces méthodes est évaluée vis-à-vis de leur capacité à modéliser la phase transitoireentre endommagement diffus (mécanique des milieux continus) et propagation defissure (mécanique de la rupture). Ces travaux sont réalisés dans le contexte dematériaux ductiles. D’après les analyses réalisées, la combinaison du modèle de ’discontinuité fortecohésive’ et la X-FEM semble être la plus prometteuse des trois approches étudiéespour allier physique et numérique. Le développement d’un tel modèle est discutéen détail. Enfin, deux critères supplémentaires sont définis : le premier pour lepassage de l’endommagement diffus au modèle de bande cohésive et un deuxièmepour le passage du modèle de bande cohésive à la rupture
The present work aims at numerically predicting the current residual strengthof large engineering structures made of ductile metals regarding accidental events,e.g. ships collision or bird strike in aviation, which may potentially lead to failure.With this aim in view, the challenge consists in reproducing within a unified finiteelement (FE)-based methodology the successive steps of micro-voiding-induceddamage, strain localization and crack propagation, if any.A key ingredient for a predictive ductile fracture model is the proper numericaltreatment of the critical transition phase of damage-induced strain localizationinside a narrow band. For this purpose, three different viewpoints in terms ofdisplacement field across the localization band are proposed involving a strong,weak and (non-linearly) regularized discontinuity, respectively.A consistent variational framework is elaborated for each of the three methods,whereby the enriched kinematics is embedded into the FE formulation using theeXtended FEM. Then, within a comparative procedure, the performance of thesemethods is assessed regarding their ability of modeling the transition phase betweendiffuse damage (continuum mechanics framework) and crack propagation (fracturemechanics framework), always in the context of ductile materials.According to the aforementioned analyses, the combination of the strong discontinuitycohesive model and the X-FEM appears to be the most promising of thethree studied approaches to bring together physics and numerics. The developmentof such a model is discussed in detail. Finally, two supplementary criteria aredefined: the first one for the passage from diffuse damage to the cohesive bandmodel and the second one for the passage from the cohesive band model to thecrack

This thesis investigates the fatigue behaviour of fibre-reinforced composite laminates. Fatigue of composite materials is a complex subject both from an experimental and numerical modelling point of view. In this context, we designed a quite extensive experimental campaign to investigate the failure mechanisms that occur when the composite material undergoes fatigue. Then, we developed a constitutive model to anticipate the mechanical behaviour and the final strength of the material. This model was implemented inside a finite element code in order to analyse the behaviour of composite structures under different loading scenarios. The final objective of this type of numerical tool is to reduce the number of test and time required for certification of aerospace composite parts
Aquesta tesi investiga la resposta a fatiga de laminats compòsits fabricats amb fibra de carboni. L'estudi de la fatiga en materials compòsits és un tema de gran complexitat tant des d'un punt de vista experimental com de modelització computacional. En aquest context, s'ha dissenyat una àmplia campanya experimental per investigar els mecanismes de dany que apareixen quan el material es sotmet a càrregues cícliques o a fatiga. Al mateix temps, s'ha desenvolupat un model constitutiu per anticipar la resposta estructural i la resistència final del material. Aquest model s'ha implementat dins d'un codi d'elements finits per tal d'analitzar estructures aeronàtutiques en diferent condicions de càrrega. L'objectiu final d'aquest models computacionals és la de reduir el nombre d'assajos experimentals i el temps que es necessita per certificar estructures aeronàutiques

Zyklische Kohäsivzonenmodelle beschreiben irreversibles Separationsverhalten und Schädigungsakkumulation unter zyklischer Belastung. In der vorliegenden Arbeit wird die Formulierung zyklischer Kohäsivzonenmodelle systematisiert und ihr Potenzial zur Simulation von Ermüdungsvorgängen analysiert. Die Kohäsivspannungs-Separations-Beziehungen werden auf Basis etablierter thermodynamischer Konzepte der Schädigungsmechanik aufgestellt. Zyklische Schädigungsakkumulation wird über die Entwicklungsgleichung der Schädigungsvariablen unter Berücksichtigung einer zustandsabhängigen Dauerfestigkeit beschrieben. Das Kohäsivzonenmodell wird erfolgreich für die Simulation von Werkstoffermüdung angewandt. Numerisch mithilfe der Methode der finiten Elemente erzeugte Rissfortschrittskurven bilden das experimentell beobachtete Ermüdungsrisswachstumsverhalten in allen Bereichen ab. Über Parameterstudien wird der Einfluss der einzelnen Modellparameter ermittelt. Darüber hinaus wird die Anwendung des zyklischen Kohäsivzonenmodells auf die Simulation von Wöhler-Versuchen vorgestellt und der Probengrößeneffekt auf das Ermüdungsverhalten untersucht. Der Zusammenhang zwischen den lokalen Beanspruchungszuständen in der Kohäsivzone und dem vorhergesagten globalen Versagensverhalten wird aufgeklärt. Die gewonnenen Erkenntnisse bilden die Grundlage für ein Konzept zur Identifikation der Kohäsivparameter, das auf der Auswertung von Wöhler- und Rissfortschrittskurven beruht
Cyclic cohesive zone models describe irreversible separation behaviour and damage accumulation under cyclic loading. In the present thesis, the formulation of cyclic cohesive zone models is systemised and their potential to simulate fatigue processes is analysed. The relation between traction and separation is described based on established thermodynamical concepts of damage mechanics. Cyclic damage accumulation is controlled by a damage evolution equation taking into account a state-dependent endurance limit. The cohesive zone model is applied successfully to the simulation of material fatigue. Fatigue crack growth rate curves, which were obtained numerically by means of the finite element method, reproduce the experimentally observed behaviour in all stages. The influences of the particular parameters of the model are determined by parametric studies. In addition, simulations of uniaxial fatigue tests using the cyclic cohesive zone model are presented. Furthermore, the size effect on the fatigue behaviour is investigated. The relation between the local states within the cohesive zone and the predicted global failure modes is explained. These findings form the foundation for a concept of parameter identification which bases on the evaluation of Wöhler-curves and fatigue crack growth rate curves

La prédiction de l’état de fissuration est un enjeu crucial pour l’analyse des structures en béton armé, qui nécessite le recours à la modélisation et à la simulation numérique. Le calcul par éléments finis des structures en béton armé pose au moins deux problèmes majeurs :d’une part il existe peu de modèles permettant de traiter à la fois l’initiation, la propagation et l’ouverture des fissures, d’autre part le diamètre généralement faible des armatures métalliques par rapport aux dimensions des structures étudiées nécessite des maillages particulièrement fins. On propose donc des solutions à ces deux problématiques basées sur l’utilisation de fonctions de niveau (level set). L’endommagement et la fissuration du béton sont modélisés à l’aide de l’approche TLS (Thick Level Set). Cette méthode,développée en tant que méthode de régularisation des modèles d’endommagement locaux, utilise une level set afin d’introduire une longueur caractéristique. Cela permet de rendre aisée la localisation de la position des fissures, et donc d’enrichir le champ de déplacement parla méthode des éléments finis étendus (X-FEM) afin de modéliser l’ouverture des macro-fissures. Concernant la modélisation des armatures, une nouvelle approche multidimensionnelle est proposée. Une représentation volumique des armatures par la méthode X-FEM est utilisée dans les zones d’intérêt afin d’obtenir des résultats précis tout en simplifiant la procédure de maillage, tandis qu’une représentation linéique est utilisée dans le reste de la structure afin de réduire le nombre de degrés de liberté du calcul. La méthode de transition développée ici permet d’assurer la cohérence des résultats obtenus
Prediction of cracking is a key point for the analysis ofreinforced concrete structures, which requires the use of Modeling and numerical simulation. The analysis of reinforced concrete structures using the finite element method raises two issues: on one hand, few models areable to deal with the initiation, the propagation and the opening of cracks, on the other hand the diameter of thereinforcements which is usually small compared to the dimensions of the structures necessitates very fine meshes. Some solutions to these two problematics areproposed, based on the use of level set functions.Damage and cracking of concrete are modeled using theThick Level Set (TLS) approach. This method,developped as a mean to regularize local damagemodels, uses a level set to introduce a characteristic length. It makes the location of the cracks easy, whichallows to enrich the displacement field with the eXtendedFinite Element Method (X-FEM) in order to model the macro-cracks opening. Concerning the modeling of thereinforcements, a new multidimensionnal approach isproposed. A volumic representation of the reinforcements with the X-FEM method is used in the zones of interest to get accurate results while simplifying the meshing process, whereas a lineic representation isused elsewhere to decrease the number of degrees of freedom. The developed transition method insures the consistency of the results

Indonesia is a developing country that suffers from earthquakes and windstorms and where at least 60% of houses are non-engineered structures, built by unskilled workers using masonry and timber. The non-engineered housing units developed in urban region are also vulnerable to seismic hazard due to the use of low quality of material and constructions method. Those structures are not resistant to extreme lateral loads or ground movement and their failure during an earthquake or storm can lead to significant loss of life. This thesis is concerned with the structural performance of Indonesian low-rise buildings made of masonry and timber under lateral seismic load. The research presented includes a survey of forms of building structure and experimental, analytical and numerical work to predict the behaviour of masonry wall and traditional timber frame buildings. Experimental testing of both masonry and timber have been carried out in Indonesia to establish the quality of materials and to provide material properties for numerical simulations. The experimental study found that the strength of Indonesia-Bali clay brick masonry are below the minimum standard required for masonry structures built in seismic regions, being at least 50% lower than the requirement specified in British Standard and Eurocode-6 (BS EN 1996-1-1:2005). In contrast, Indonesian timber materials meet the strength classes specified in British Standard/Eurocode- 5 (BS EN 338:2009) in the range of strength grade D35-40 and C35).Structural tests under monotonic and cyclic loading have been conducted on building components in Indonesia, to determine the load-displacement capacity of local hand-made masonry wall panels and timber frames in order to: (1) evaluate the performance of masonry and timber frame structure, (2) investigate the dynamic behaviour of both structures, (3) observe the effect of in-plane stiffness and ductility level, and (4) examine the anchoring joint at the base of timber frame that resists the overturning moment. From these tests, the structural ductility was found to be less than two which is below the requirement of the relevant guidelines from the Federal Emergency Management Agency, USA (FEMA-306). It was also observed that the lateral stiffness of masonry wall is much higher than the equivalent timber frame of the same height and length. The experimental value of stiffness of the masonry wall panel was found to be one-twelfth of the recommended values given in FEMA-356 and the Canadian Building code. The masonry wall provides relatively low displacement compared to the large displacement of the timber frame at the full capacity level of lateral load, with structural framing members of the latter remaining intact. The weak point of the timber frame is the mechanical joint and the capacity of slip joint governs the lateral load capacity of the whole frame. Detailed numerical models of the experimental specimens were setup in Abaqus using three-dimensional solid elements. Cohesive elements were used to simulate the mortar behaviour, exhibiting cracking and the associated physical separation of the elements. Appropriate contact definitions were used where relevant, especially for the timber frame joints. A range of available material plasticity models were reviewed: Drucker-Prager, Crystalline Plasticity, and Cohesive Damage model. It was found that the combination of Crystalline Plasticity model for the brick unit and timber, and the Cohesive Damage model for the mortar is capable of simulating the experimental load-displacement behaviour fairly accurately. The validated numerical models have been used to (1) predict the lateral load capacity, (2) determine the cracking load and patterns, (3) carry out a detailed parametric study by changing the geometric and material properties different to the experimental specimens. The numerical models were used to assess different strengthening measures such as using bamboo as reinforcement in the masonry walls for a complete single storey, and a two-storey houses including openings for doors and windows. The traditional footing of the timber structures was analysed using Abaqus and was found to be an excellent base isolation system which partly explains the survival of those structures in the past earthquakes. The experimental and numerical results have finally been used to develop a design guideline for new construction as well as recommendations for retrofitting of existing structures for improved performance under seismic lateral load.

The focus of the thesis is on developing new damage models for brittle materials and using these to study delamination of composite structures. Chapter 1 gives an introductory literature review in order to motivate the work undertaken in the chapters to follow. Chapter 2 deals with a new micropolar damage model for delamination in composites. By combining phase field theory and peridynamics, Chapter 3 develops a new formalism to study damage in brittle materials under dynamic loading. Chapter 4 exploits and extends this idea for modelling delamination of composites. An extended chapter-wise summary of the contributions in the thesis is provided below. In Chapter 2, a micropolar cohesive damage model for delamination of composites is proposed. The main idea is to embed micropolarity, which brings in an added layer of kinematics through the micro-rotation degrees of freedom within a continuum model to account for the micro-structural effects during delamination. The resulting cohesive model, described through a modified traction separation law, includes micro-rotational jumps in addition to displacement jumps across the interface. The incorporation of micro-rotation requires the model to be supplemented with physically relevant material length scale parameters, whose effects during delamination in modes I and II are brought forth using numerical simulations appropriately supported by experimental evidences. In Chapter 3, we attempt at reformulating the phase field theory within the framework of peridynamics (PD) to arrive at a non-local continuum damage model. This obtains a better criterion for bond breaking in PD, marking a departure from the inherently ad-hoc bond-stretch-based or bond-energy-based conditions and thus allowing the model to simulate fragmentation which a phase field model cannot by itself accomplish. Moreover, posed within the PD setup, the integral equation for the phase field eases the smoothness restrictions on the field variable. Taking advantages of both the worlds, the scheme thus offers a better computational approach to problems involving cracks or discontinuities. Starting with Hamilton’s principle, an equation of the Ginzburg-Landau type with dissipative correction is arrived at as a model for the phase field evolution. A constitutive correspondence route is followed to incorporate classical constitutive relations within our PD model. Numerical simulations of dynamic crack propagation (including branching) and the Kalthoff-Winkler experiment are also provided. To demonstrate the natural ability of the model to prevent interpenetration, a mode II delamination simulation is presented. A brief discussion on the convergence of PD equations to classical theory is provided in the Appendix B. In Chapter 4, we extend and exploit the phase field based PD damage model, developed in Chapter 3, for studying delamination of composites. Utilizing the phase field augmented PD framework, our idea is to model the interfacial cohesive damage through degradation functions and the fracture or fragmentation through the critical energy release rate. Our model eliminates the conventional traction-separation law (TSL) that is known to result in the popular cohesive zone model (CZM). In the process, the approach potentially addresses some limitations of the existing techniques, which make use of an empirical interaction among different modes of loading (e.g. mode I, mode II etc.). By regarding delamination under different loading conditions as problems that differ only in their boundary conditions, our approach provides for a more general scheme for tracking delamination. Our proposal thus accords no special treatment to the different modes and can handle general spatial locations of weaker interface layers. With no special crack tracking algorithms or additional ad-hoc criteria for crack propagation, considerable computational simplicity also accrues. The approach can tackle cases where cracks may propagate even in the bulk material body. The new bond breaking criterion that we employ replaces the ad-hocism inherent in bond-stretch-based or bond-energy-based conditions. Using numerical simulations on mode I (double cantilever beam test), mode II (end loaded split and end notched flexure tests) and mixed mode (fixed ratio mixed mode test) delamination cases, the model is validated against relevant experimental observations. Simulations on modified mixed mode bending test and multiple layer delamination test are also presented. The thesis is wound up in Chapter 5 with a summary of accomplished research and some suggestions for future research.

abstract: Cohesive zone model is one of the most widely used model for fracture analysis, but still remains open ended field for research. The earlier works using the cohesive zone model and Extended finite element analysis (XFEM) have been briefly introduced followed by an elaborate elucidation of the same concepts. Cohesive zone model in conjugation with XFEM is used for analysis in static condition in order to check its applicability in failure analysis. A real time setup of pipeline failure due to impingement is analyzed along with a detailed parametric study to understand the influence of the prominent design variable. After verifying its good applicability, a creep model is built for analysis where the cohesive zone model with XFEM is used for a time dependent creep loading. The challenge in this simulation was to achieve coupled behavior of cracks initiation and propagation along with creep loading. By using Design of Experiment, the results from numerical simulation were used to build an equation for life prediction for creep loading condition. The work was further extended to account for fatigue damage accumulation for high cycle fatigue loading in cohesive elements. A model was conceived to account for damage due to fatigue loading along within cohesive zone model for cohesive elements in ABAQUS simulation software. The model was verified by comparing numerical modelling of Double cantilever beam under high cycle fatigue loading and experiment results from literature. The model was also applied to a major industrial problem of blistering in Cured-In-Plane liner pipelines and a demonstration of its failure is shown. In conclusion, various models built on cohesive zone to address static and time dependent loading with real time scenarios and future scope of work in this field is discussed.
Dissertation/Thesis
Masters Thesis Mechanical Engineering 2016

Στην παρούσα διπλωματική εργασία αναπτύχθηκε τρισδιάστατο αριθμητικό μοντέλο με βάση την μέθοδο των πεπερασμένων στοιχείων για την προσομοίωση της μηχανικής συμπεριφοράς συνδέσεων με κόλλα πολύστρωτων πλακών. Το μοντέλο αναπτύχθηκε χρησιμοποιώντας το εμπορικό πακέτο πεπερασμένων στοιχείων ANSYS. Για την προσομοίωση της συμπεριφοράς της κόλλας χρησιμοποιήθηκαν δύο προσεγγίσεις: η μοντελοποίηση της ζώνης συνοχής και η μοντελοποίηση της βλάβης του συνεχούς μέσου. Οι δύο αυτές προσεγγίσεις συγκρίθηκαν τόσο ως προς την αξιοπιστία τους, η οποία καθορίζεται από την σύγκριση με πειραματικά αποτελέσματα, όσο και ως προς την ευκολία εφαρμογής τους, η οποία καθορίζεται από τα δεδομένα που απαιτούνται και τον υπολογιστικό χρόνο. Η σύγκριση των δύο μεθοδολογιών έγινε στην βάση της εφαρμογής τους για την προσομοίωση της μηχανικής συμπεριφοράς σε φόρτιση τύπου Ι σύνδεσης με κόλλα μεταξύ δύο ψευδοισότροπων CFRP πολύστρωτων πλακών. Το συγκεκριμένο πρόβλημα επελέγη διότι υπήρχαν διαθέσιμα πειραματικά αποτελέσματα προς σύγκριση στο Εργαστήριο. Οι πολύστρωτες πλάκες μοντελοποιήθηκαν χρησιμοποιώντας το στρωματικό στοιχείο του ANSYS SOLID185. Στο στοιχείο αυτό κάθε στρώση μοντελοποιείται ξεχωριστά ως ορθότροπο υλικό. Η εφαρμογή της μοντελοποίησης της ζώνης συνοχής έγινε μέσω της χρήσης του στοιχείου του ANSYS INTER205. Για την εφαρμογή της μοντελοποίησης της βλάβης του συνεχούς μέσου αναπτύχθηκε μακρο-ρουτίνα χρησιμοποιώντας την γλώσσα προγραμματισμού του κώδικα ANSYS. Τα αριθμητικά αποτελέσματα έδειξαν ότι και οι δύο μεθοδολογίες προσομοιώνουν με ικανοποιητική ακρίβεια την καμπύλη δύναμης-μετατόπισης της σύνδεσης. Σχετικά με την ευκολία εφαρμογής των δύο μεθόδων, η σύγκριση έδειξε ότι η μέθοδος της μοντελοποίησης της ζώνης συνοχής υπερτερεί έναντι της μεθόδου μοντελοποίησης της βλάβης του συνεχούς μέσου διότι απαιτεί μικρότερο αριθμό δεδομένων, μειονεκτεί όμως ως προς τον απαιτούμενο υπολογιστικό χρόνο. Και οι δύο μέθοδοι κρίνονται κατάλληλες για χρήση στην αριθμητική σχεδίαση συνδέσεων με κόλλα.
This work is based on the development of three-dimensional numerical model based on the finite element method to simulate the mechanical behavior of adhesive bonded joints in composite materials. The model was developed in finite element procedures under the framework of the commercial software ANSYS. To simulate the behavior of the adhesive used two approaches: the cohesive zone modeling (CZM) and the continuum damage modeling (CDM). These two approaches are compared both in terms of reliability, which is determined by comparison with experimental results, and applicability, which is determined by the parameters required and the computational time. The comparison between the two methodologies was the basis of their application to simulate the mechanical behavior under mode-I fracture behavior of adhesively bonded joints between two CFRP plates. This problem was chosen because there were experimental results to compare in the laboratory. The sandwich plates are modeled using the stromal element of ANSYS SOLID185. This item each layer separately modeled as orthotropic material. The adhesive is modeled using the interface element of ANSYS INTER205. For the purpose of modeling the failure of continuous medium developed macro routine using the programming language code ANSYS. The numerical results showed that both methodologies simulate sufficient precision the curve force-displacement of the connection. About applicability of the two methods, the comparison showed that the process of cohesive zone modeling outweighs the process of continuum damage modeling because it requires less number of parameters, but falls to the computational time. Both methods are suitable for use in numerical design of adhesively bonded joints.

Zyklische Kohäsivzonenmodelle beschreiben irreversibles Separationsverhalten und Schädigungsakkumulation unter zyklischer Belastung. In der vorliegenden Arbeit wird die Formulierung zyklischer Kohäsivzonenmodelle systematisiert und ihr Potenzial zur Simulation von Ermüdungsvorgängen analysiert. Die Kohäsivspannungs-Separations-Beziehungen werden auf Basis etablierter thermodynamischer Konzepte der Schädigungsmechanik aufgestellt. Zyklische Schädigungsakkumulation wird über die Entwicklungsgleichung der Schädigungsvariablen unter Berücksichtigung einer zustandsabhängigen Dauerfestigkeit beschrieben. Das Kohäsivzonenmodell wird erfolgreich für die Simulation von Werkstoffermüdung angewandt. Numerisch mithilfe der Methode der finiten Elemente erzeugte Rissfortschrittskurven bilden das experimentell beobachtete Ermüdungsrisswachstumsverhalten in allen Bereichen ab. Über Parameterstudien wird der Einfluss der einzelnen Modellparameter ermittelt. Darüber hinaus wird die Anwendung des zyklischen Kohäsivzonenmodells auf die Simulation von Wöhler-Versuchen vorgestellt und der Probengrößeneffekt auf das Ermüdungsverhalten untersucht. Der Zusammenhang zwischen den lokalen Beanspruchungszuständen in der Kohäsivzone und dem vorhergesagten globalen Versagensverhalten wird aufgeklärt. Die gewonnenen Erkenntnisse bilden die Grundlage für ein Konzept zur Identifikation der Kohäsivparameter, das auf der Auswertung von Wöhler- und Rissfortschrittskurven beruht.
Cyclic cohesive zone models describe irreversible separation behaviour and damage accumulation under cyclic loading. In the present thesis, the formulation of cyclic cohesive zone models is systemised and their potential to simulate fatigue processes is analysed. The relation between traction and separation is described based on established thermodynamical concepts of damage mechanics. Cyclic damage accumulation is controlled by a damage evolution equation taking into account a state-dependent endurance limit. The cohesive zone model is applied successfully to the simulation of material fatigue. Fatigue crack growth rate curves, which were obtained numerically by means of the finite element method, reproduce the experimentally observed behaviour in all stages. The influences of the particular parameters of the model are determined by parametric studies. In addition, simulations of uniaxial fatigue tests using the cyclic cohesive zone model are presented. Furthermore, the size effect on the fatigue behaviour is investigated. The relation between the local states within the cohesive zone and the predicted global failure modes is explained. These findings form the foundation for a concept of parameter identification which bases on the evaluation of Wöhler-curves and fatigue crack growth rate curves.

In der vorliegenden Arbeit wird die Frage untersucht, ob effektive Eigenschaften von Verbunden auch nach dem Auftreten einer Dehnungslokalisierung aufgrund von entfestigendem Materialverhalten noch durch numerische Homogenisierungsmethoden berechnet werden können. Ihr Nutzen für diesen Anwendungsfall wird in der Literatur kritisch beurteilt. Aus diesem Grund werden hier systematisch alle Teilaufgaben betrachtet, die zu diesem Zweck gelöst werden müssen. Die erste dieser Aufgaben ist die Charakterisierung der einzelnen Verbundbestandteile. Zur Demonstration einer experimentell gestützten Charakterisierung wird ein glasfaserverstärktes Epoxidharz als Beispielmaterial gewählt. Neben der Beschreibung von Faser- und Matrixmaterial wird besonderes Augenmerk auf die Charakterisierung der Grenzschicht zwischen beiden gelegt. Die für die Glasfasern vorliegenden Festigkeitsmessungen entsprechen nicht der Kettenhypothese. Daher werden zahlreiche Verallgemeinerungen der Weibull-Verteilung untersucht, um störende Effekte zu erfassen. Schließlich werden Wahrscheinlichkeitsverteilungen hergeleitet, die Faserbrüche im Bereich der Einspannung einbeziehen. Die Messwerte können von diesen Verteilungen gut wiedergegeben werden. Zusätzlich macht ihre Anwendung das aufwändige Aussortieren und Wiederholen jener Experimente unnötig, bei denen der Faserbruch im Klemmbereich auftritt. Zur Modellierung der Grenzfläche wird ein Kohäsivzonengesetz entwickelt. Die Bestimmung seiner Parameter erfolgt anhand von Daten aus Pullout- und Einzelfaserfragmentierungsversuchen. Aus diesen ermittelte Festigkeiten und Energiefreisetzungsraten weisen eine sehr gute Übereinstimmung zwischen beiden Versuchen auf. Dabei erfolgt die Parameteridentifikation mithilfe von Finite-Elemente-Modellen anstatt der häufig genutzten vereinfachten analytischen Modelle, welche üblicherweise eine schlechtere Übereinstimmung erreichen. Sobald eine Dehnungslokalisierung auftritt, ist neben der Materialmodellierung auch das Homogenisierungsschema zu verallgemeinern. Zu diesem gehören die Generierung repräsentativer Volumenelemente, Randbedingungen (RB) und ein Mittelungsoperator. Anhand des aktuellen Standes der Literatur werden die Randbedingungen als ein signifikanter Schwachpunkt von Homogenisierungsverfahren erkannt. Daher erfolgt die Untersuchung periodischer RB, linearer Verschiebungsrandbedingungen und minimal kinematischer RB sowie zweier adaptiver RB, nämlich Lokalisierungspfad-ausgerichteter RB und generalisiert periodischer RB. Unter der Bezeichnung Tesselationsrandbedingungen wird ein weiterer Typ adaptiver RB vorgeschlagen. Zunächst erfolgt der Beweis, dass alle drei adaptiven RB die Hill-Mandel-Bedingung erfüllen. Des Weiteren wird mittels einer Modifikation der Hough-Transformation ein systematischer Fehler derselben bei der Bestimmung der Richtung von Lokalisierungszonen eliminiert. Schließlich werden die Eigenschaften aller Randbedingungen an verschiedenen Beispielen demonstriert. Dabei zeigt sich, dass nur Tesselationsrandbedingungen sowohl beliebige Richtungen von Lokalisierungszonen erlauben als auch fehlerhafte Lokalisierungen in Eckbereichen ausschließen. Zusammengefasst können in der Literatur geäußerte grundlegende Einschränkungen hinsichtlich der Anwendbarkeit numerischer Homogenisierungsverfahren beim Auftreten von Dehnungslokalisierungen aufgehoben werden. Homogenisierungsmethoden sind somit auch für entfestigendes Materialverhalten anwendbar.
The thesis at hand is concerned with the question if numerical homogenization schemes can be of use in deriving effective material properties of composite materials after the onset of strain localization due to strain softening. In this case, the usefulness of computational homogenization methods has been questioned in the literature. Hence, all the subtasks to be solved in order to provide a successful homogenization scheme are investigated herein. The first of those tasks is the characterization of the constituents, which form the composite. To allow for an experimentally based characterization an exemplary composite has to be chosen, which herein is a glass fiber reinforced epoxy. Hence the constituents to be characterized are the epoxy and the glass fibers. Furthermore, special attention is paid to the characterization of the interface between both materials. In case of the glass fibers, the measured strength values do not comply with the weakest link hypothesis. Numerous generalizations of the Weibull distribution are investigated, to account for interfering effects. Finally, distributions are derived, that incorporate the possibility of failure inside the clamped fiber length. Application of such a distribution may represent the measured data quite well. Additionally, it renders the cumbersome process of sorting out and repeating those tests unnecessary, where the fiber fails inside the clamps. Identifying the interface parameters of the proposed cohesive zone model relies on data from pullout and single fiber fragmentation tests. The agreement of both experiments in terms of interface strength and energy release rate is very good, where the parameters are identified by means of an evaluation based on finite element models. Also, the agreement achieved is much better than the one typically reached by an evaluation based on simplified analytical models. Beside the derivation of parameterized material models as an input, the homogenization scheme itself needs to be generalized after the onset of strain localization. In an assessment of the current state of the literature, prior to the generation of representative volume elements and the averaging operator, the boundary conditions (BC) are identified as a significant issue of such a homogenization scheme. Hence, periodic BC, linear displacement BC and minimal kinematic BC as well as two adaptive BC, namely percolation path aligned BC and generalized periodic BC are investigated. Furthermore, a third type of adaptive BC is proposed, which is called tesselation BC. Firstly, the three adaptive BC are proven to fulfill the Hill-Mandel condition. Secondly, by modifying the Hough transformation an unbiased criterion to determine the direction of the localization zone is given, which is necessary for adaptive BC. Thirdly, the properties of all the BC are demonstrated in several examples. These show that tesselation BC are the only type, that allows for arbitrary directions of localization zones, yet is totally unsusceptible to spurious localization zones in corners of representative volume elements. Altogether, fundamental objections, that have been raised in the literature against the application of homogenization in situations with strain localization, are rebutted in this thesis. Hence, the basic feasibility of homogenization schemes even in case of strain softening material behavior is shown.