The continuous increase of the voltage levels in power transmission systems has lead to the occurrence of higher switching transients during their operation. The design of equipment and grid components able to sustain such a stressful operation, requires an intensive study of the electric field stress generated by these transients, and their distribution to the vicinity of each configuration. Sphere-plane gaps are the most theoretically and practically interesting electrode configurations. So far, the majority of the conducted work is referred to the study of the discharge characteristics of this structure. However, a study of the electrostatic electric field is required. An accurate calculation of the electric field can contribute significantly to an even better understanding of the discharge characteristics and the principles behind them. In this project, is presented a simulation approach for the calculation of the electro-static field of a sphere-plane configuration, varying the dimensions of the sphere and the gap distance. For this purpose, a Finite Element Method (FEM) solver was used, in which the configuration was designed and the numerical solution of the problem was implemented. After that, an attempt was performed to specify the breakdown voltage based on the electric field calculation and distribution. Useful results were recorded from both the simulation of the electrostatic model and the calculation of the breakdown voltage. One of the most important findings, was the specification of an approximate relation between the diameters of the sphere and the tube where this is mounted. As a consequence, the study of the electric field distribution became easier, while at the same time an accurate calculation of the breakdown voltage was achieved. A series of validations were performed, through the comparison with the already ex-isting, published and unpublished, experimental tests and a number of conclusions were listed. One of the most significant, was the specification of the correlation between the electrostatic model and test measurements and how these different approaches can be linked to each other in a practically efficient way. At the end, there is a proposal for further work on the subject, and possible improvements of the already conducted work. / Ökningen av spänningsnivåerna i kraftöverföringssystem har lett till högre kop-plingstransienter under drift. Konstruktionen av utrustning och nätverkskomponen-ter som kan motstå en sådan påkänning, kräver en noggrann förstudie av de elektriska fältet som genereras av dessa transienter. Ett gap av sfär-platta är den mest teoretiskt och praktiskt intressanta elektrodkon-figurationen att studera. Hittills har majoritetet av genomfört arbete avsett att studera urladdningsegenskaper för denna struktur. Dock krävs studie av elektrostatiska elek-triska fältet. En noggrann beräkning av elektriska fältet kan bidra till en ännu bättre förståelse för urladdningsegenskaper och principerna bakom dem. I detta projekt presenteras en simuleringsmetod för beräkning av elektrostatiska fäl-tet av en sfär-plan konfiguration, med varierande dimensioner av sfären och gapavstån-det. För detta ändamål har använts en Finite Element Method (FEM) lösning, där kon-figurationen utformades och problemets numeriska lösningen genomfördes. Därefter har gjorts ett försök för att ange genombrottsspänningen baserad på beräkning och distribution av elektriskta fältet. Resultat registrerades från både simulering av elektrostatiska modellen och beräkn-ing av genomslagsspänningen. Ett av de viktigaste resultaten var specifikationen av ett ungefärligt förhållande mellan diametrarna av sfären och röret där sfären är monterad. Som en konsekvens blev studiet av elektriska fältfördelningen lättare, medan en exakt beräkning av genomslagsspänningen uppnåddes. En valideringsserie har genomförts genom jämförelse med de redan existerande, publicerade och opublicerade, experimentella tester och ett antal slutsatser har noterats. En av de mest meningsfulla, var specifikation av sambandet mellan den elektrostatiska modellen och provmätningarna samt hur dessa olika tillvägagångssätt kan kopplas till varandra på ett praktiskt och effektivt sätt. Slutligen finns det ett förslag för fortsatta arbete samt eventuella förbättringar av redan genomfört arbete.

Les Surfaces à Haute Impédance (SHI) ont été largement étudiées pour améliorer toutes sortes de performances des antennes, comme le gain, le facteur de qualité, les formes et dimensions. L'objectif de cette thèse est de modéliser les structures de SHI et de caractériser leurs performances en vue de futures applications aux antennes.Après une brève introduction aux structures SHI et une étude de quelques modèles analytiques fréquemment traités dans la littérature, deux nouvelles méthodes numériques sont proposées pour calculer l'impédance de surface de structures SHI. Ces deux méthodes (dites « méthode du flux de Poynting » et « méthode <E>/<H> ») sont validées sur des structures symétriques, puis mises en service sur des structures de SHI asymétriques. Elles sont également validées par comparaison de résultats analytiques, numériques et expérimentaux.Nous présentons ensuite un modèle équivalent basé sur l'idée de remplacer les structures hétérogènes de SHI par une surface homogène, caractérisée par son impédance surfacique. Ce modèle nous permet d'avoir une prédiction avec un temps de calcul et une occupation de mémoire PC largement réduits. / High impedance surfaces (HIS) have proved good candidates for antenna miniaturization or antennas performance improvement. Within a certain frequency band, they can enhance the gain of an antenna while simultaneously suppressing the unwanted surface waves.In this thesis, the focus of our work is on numerical modeling of these structures by using the finite element method (FEM) based on edge elements.One of our contributions is that we propose two new numerical methods (the Poynting flux method and <E>/<H> method) to calculate the surface impedance not only for HIS structures with symmetric geometries, but also for those with asymmetric geometries. These two numerical methods have been validated by comparing analytical, numerical and experimental results.Another significant contribution of the thesis is that we introduce an equivalent model, based on the idea of replacing the heterogeneous HIS structures by a homogeneous surface, characterized by its surface impedance. Compared with the normal model, this equivalent model can save computing time and memory space.

Ces travaux s'inscrivent dans le cadre des recherches menées dans le cadre d'une collaboration entre le laboratoire Roberval de l'Université de Technologie de Compiègne et l'équipe dans le cadre du projet ANR-14-CE07-0035 LASMIS de l'Institut Charles Delaunay de l'Université de Technologie de Troyes. Nous présentons dans ces travaux une h-méthodologie adaptative tridimensionnelle des éléments finis afin de représenter l'initiation et la propagation des fissures dans des matériaux ductiles. Un modèle élasto-plastique couplé à l'endommagement isotrope proposé par l'équipe du LASMIS/UTT est utilisé. Les applications visées à terme concernent principalement la mise en forme des métaux. Dans ce contexte, une formulation Lagrangienne actualisée est employée et des remaillages fréquents s'avèrent essentiels afin d'une part d'éviter les fortes distorsions d'éléments dues aux grandes déformations plastiques et d'autre part de suivre les modifications de la topologie résultant de la création de fissures. La taille du nouveau maillage doit permettre à moindre coût représenter avec précision l'évolution des gradients des quantités physiques représentatives des phénomènes étudiées (plasticité, endommagement...). Nous proposons des indicateurs empiriques de taille d'éléments basés sur la déformation plastique ainsi que sur l'endommagement. Une courbe définie par morceau représente l'évolution de la taille d'élément suivant la sévérité de la plasticité et le cas échéant de l'endommagement. Les fissures sont représentées par une méthode de destruction d'éléments qui permet une description aisée de la géométrie de ces dernières et une gestion simplifiée de la fissuration sans nul besoin de critères additionnels. En revanche, pour permettre une description réaliste des fissures, ces dernières doivent être représentées par l'érosion des éléments de plus petite taille. Un solveur ABAQUS/Explicit® est utilisé avec des éléments tétraédriques quadratiques (C3D10M) évitant notamment les problèmes de verrouillage numérique survenant lors de l'analyse de structures en matériau compressible ou quasi-incompressible. Le contrôle de la plus petite taille du maillage est important dans un contexte explicite. De surcroît, pour les phénomènes adoucissant, la solution dépend de la taille de maille considérée alors comme un paramètre intrinsèque. Une étude nous a permis de constater que lorsque le maillage est suffisamment raffiné, les effets de la dépendance au maillage se réduisaient. Dans la littérature, les coûts de maillage ou de remaillage fréquents sont souvent considérés comme prohibitifs et de nombreux auteurs s'appuient sur cet argument pour introduire, avec succès certes, des méthodes alternatives qui limitent le coût des opérations de remaillage sans toutefois les éliminer (XFEM par exemple). Nos travaux montrent que le coût d'un remaillage local est négligeable par rapport au calcul. Compte tenu de la complexité de la géométrie et de la nécessité de raffiner le maillage, la seule alternative à ce jour est d'utiliser un mailleur en tétraèdres. La stratégie de remaillage local en tétraèdre s'appuie sur une méthode de bisection suivie si nécessaire d'une optimisation locale du maillage proposé par A. Rassineux en 2003. Le remaillage, même local, doit s'accompagner de procédures de transfert de champ des variables nodales et aux points d'intégration. Les variables nodales sont, comme le fait la plupart des auteurs, transférées en utilisant les fonctions de forme éléments finis. Le transfert de champ en 3D aux points de Gauss et les nombreux problèmes sous-jacents ont été relativement peu abordés dans la littérature. / This work is part of the research carried out in the framework of a collaboration between the Roberval laboratory of the Compiègne University of Technology and the team within the framework of the project ANR-14-CE07-0035 LASMIS of the Charles Delaunay Institute of Technology University of Troyes. In this work, we present a three-dimensional adaptive Pi-methodology of finite elements to represent the initiation and propagation of cracks in ductile materials. An elastoplastic model coupled with the isotropic damage proposed by the LASMIS / UTT team is used. The targeted applications will mainly concern the metal forming. In this context, an updated Lagrangian formulation is used and frequent remeshing is essential in order to avoid the strong distortion of elements due to large plastic deformations and to follow the modifications of the topology resulting in the creation of cracks. The size of the new mesh must allow at a lower cost to accurately represent the evolution of the gradients of the physical quantities representative of the studied phenomena (plasticity, damage ...). We propose empirical indicators of size of elements based on the plastic deformation as well as on the damage. A piecewise defined curve represents the evolution of the element size according to the severity of the plasticity and, if appropriate, the damage. The cracks are represented by a method of destruction of elements which allows an easy description of the geometry and a simplified treatment of the cracking without any need for additional criteria. On the other hand, to allow a realistic description of the cracks, the latter must be represented by erosion smaller elements. An ABAQUS / Explicit@ solver is used with quadratic tetrahedral elements (C3DIOM), avoiding in particular the problems of numerical locking occurring during the analysis of structures in compressible or quasi-incompressible material. The control of the smaller mesh size is important in an explicit context. In addition, for softening phenomena, the solution depends on the mesh size considered as an intrinsic parameter. A study has shown that when the mesh is sufficiently refined, the effects of mesh dependence are reduced. In the literature, the costs of frequent meshing or remeshing are often considered prohibitive and many authors rely on this argument to introduce, with success, alternative methods that limit the cost of remeshing operations without eliminating them ( XFEM for example). Our work shows that the cost of local remeshing is negligible compared to the calculation. Given the complexity of the geometry and the need to refine the mesh, the only alternative to date is to use a mesh in tetrahedra. The strategy of local remeshing tetrahedron is based on a bisection method followed if necessary by a local optimization of the grid proposed by A. Rassineux in 2003. The remeshing, even local, must be accompanied by field transfer procedures on both nodal variables and integration points. Node variables are, as most authors do, transferred using finite element shape functions. The 3D field transfer at Gauss points and the many underlying problems have been relatively untouched in the literature. The main difficulties to be solved in order to ensure the "quality" of the transfer concern the limitation of numerical diffusion, the lack of information near borders, the respect of boundary conditions, the equilibrium, the calculation costs, the filtering of the information points, crucial problems in 3D where the number of Gauss points used is several hundred. We propose a so-called "hybrid" method which consists, initially, in extrapolating the data at the Gauss points, in the nodes by diffuse interpolation and then in using the finite element form functions to obtain the value at the point considered.

Les phénomènes de couplage électromagnétique entre les lignes aé- riennes de transmission d\'énergie et des structures voisines sont inévitables, surtout dans les zones densément peuplées. Les effets indésirables découlants de cette proximité sont variés, allant de l\'établissement des tensions dangereuses à l\'apparition de phénomènes de corrosion dus au courant alternatif. L\'étude de cette classe de problèmes est nécessaire pour assurer la sécurité dans les alentours de la zone d\'interaction et aussi pour préserver l\'intégrité des équipements et des dispositifs présents. Cependant, la modélisation compl ète de ce type d\'application implique la représentation tridimensionnelle de la région d\'intérêt et nécessite des méthodes numériques de calcul de champs spécifiques. Dans ces travaux, des problèmes liés à la circulation de courants électriques dans le sol (ou de couplage dit conductif) seront abordés avec la méthode des éléments finis. D\'autres problèmes résultants de la variation temporelle des champs électromagnétiques (ou de couplage dit inductif) seront aussi considérés et traités avec la méthode PEEC (Partial Element Equivalent Circuit) généralisée. Plus précisément, une condition limite particulière sur le potentiel électrique est proposée pour tronquer le domaine de calcul dans l\'analyse par éléments finis des problèmes de couplage conductif et une formulation PEEC complète pour la modélisation de problèmes de couplage inductif est présentée. Des con gurations tests de complexités croissantes sont considérées pour valider les approches précédentes. Ces travaux visent ainsi à apporter une contribution à la modélisation de cette classe de problèmes, qui tendent à devenir communs avec l\'expansion des réseaux électriques. / Electromagnetic coupling phenomena between overhead power transmission lines and other nearby structures are inevitable, especially in densely populated areas. The undesired effects resulting from this proximity are manifold and range from the establishment of hazardous potentials to the outbreak of alternate current corrosion phenomena. The study of this class of problems is necessary for ensuring security in the vicinities of the interaction zone and also to preserve the integrity of the equipment and of the devices there present. However, the complete modeling of this type of application requires the three- -dimensional representation of the region of interest and needs specific numerical methods for field computation. In this work, the modeling of problems arising from the flow of electrical currents in the ground (the so-called conductive coupling) will be addressed with the finite element method. Those resulting from the time variation of the electromagnetic fields (the so-called inductive coupling) will be considered as well, and they will be treated with the generalized PEEC (Partial Element Equivalent Circuit) method. More specifically, a special boundary condition on the electric potential is proposed for truncating the computational domain in the finite element analysis of conductive coupling problems, and a complete PEEC formulation for modeling inductive coupling problems is presented. Test configurations of increasing complexities are considered for validating the foregoing approaches. These works aim to provide a contribution to the modeling of this class of problems, which tend to become common with the expansion of power grids.

This work has been focused on characterisation of FSW joints, and modelling of the process, both analytically and numerically. The Swedish model for final deposit of nuclear fuel waste is based on copper canisters as a corrosion barrier with an inner pressure holding insert of cast iron. Friction Stir Welding (FSW) is the method to seal the copper canister, a technique invented by The Welding Institute (TWI). The first simulations were based on Rosenthal’s analytical medium plate model. The model is simple to use, but has limitations. Finite element models (FEM) were developed, initially with a two-dimensional geometry. Due to the requirements of describing both the heat flow and the tool movement, three-dimensional models were developed. These models take into account heat transfer, material flow, and continuum mechanics. The geometries of the models are based on the simulation experiments carried out at TWI and at Swedish Nuclear Fuel Waste and Management Co (SKB). Temperature distribution, material flow and their effects on the thermal expansion were predicted for a full-scale canister and lid. The steady state solutions have been compared with temperature measurements, showing good agreement. In order to understand the material flow during welding a marker technique is used, which involves inserting dissimilar material into the weld zone before joining. Different materials are tested showing that brass rods are the most suitable material in these welds. After welding, the weld line is sliced, etched and examined by optical microscope. To understand the material flow further, and in the future predict the flow, a FEM is developed. This model and the etched samples are compared showing similar features. Furthermore, by using this model the area that is recrystallised can be predicted. The predicted area and the grain size and hardness profile agree well. Microstructure and hardness profiles have been investigated by optical microscopy, Scanning Electron Microscopy (SEM), Electron Back Scatter Diffraction (EBSD) and Rockwell hardness measurements. EBSD visualisation has been used to determine the grain size distribution and the appearance of twins and misorientation within grains. The orientation maps show a fine uniform equiaxed grain structure. The root of the weld exhibits the smallest grains and many annealing twins. The appearance of the nugget and the grain size depends on the position of the weld. A large difference can be seen both in hardness and grain size between the start of the weld and when the steady state is reached. / QC20100719

With the development of the design and manufacture technology, composite materials are widely used in the aeronautical industry. But, one of the main concerns which affects the application of composites is foreign object impact. The damages induced by the Low Velocity Impact (LVI), which can significantly reduce the strength of the structures, can’t be easily inspected routinely. The so-called Barely Visible Impact Damages (BVID) due to LVI typically includes interlaminar delamination, matrix cracks and fibre fracture at the back face. Previous researches have shown that the results of LVI test are similar to that of the Quasi-Static Load (QSL) test. The initiation and propagation of delamination can be detected more easily in the QSL test and the displacement and reaction force of the impactor can be controlled and measured much more accurately. Moreover, it is easier to model QSL tests than dynamic impacts. To investigate the impact damage induced by LVI, a Finite Element (FE) model employing cohesive elements was used. At the same time, the ply clustering effect, when several plies of the same orientation were stack together, was modelled in the FE model in terms of damage resistance and damage size. A bilinear traction-separation law was introduced in the cohesive elements employed to simulate the initiation and propagation of the impact damage and delamination. Firstly, a 2D FE model of the Double Cantilever Beam (DCB) and End Notched Flexure (ENF) specimens were built using the commercial FEM software ABAQUS. The results have shown that the cohesive elements can be used to simulate mode I and mode II delamination sufficiently and correctly. Secondly, an FE model of a composite plate under QSL but without simulating damage was built using the continuum shell elements. Agreement between the FEA results with published test results is good enough to validate the capability of continuum shell elements and cohesive elements in modelling the composite laminate under the transverse load condition (QSL). Thirdly, an FE model containing discrete interface delamination and matrix cracks at the back face of the composite plate was built by pre-setting the cohesive failure elements at potential damage locations according to the experimental observation. A cross-ply laminate was modelled first where fewer interfaces could be delaminated. Good agreement was found in terms of the delamination area and impactor’s displacement-force curve. Finally, the effect of ply clustering on impact damage resistance was studied using Quasi-Isotropic (QI) layup laminates. Because of the limited time available for calculation, the simulation was only partly completed for the quasi-isotropic laminates (L2 configuration) which have more delaminated interfaces. The results showed that cohesive elements obeying the bilinear traction-separation law were capable of predicting the reaction force in quasi-isotropic laminates. However, discrepancies with the test results in terms of delamination area were observed for quasi-isotropic laminates. These discrepancies are mainly attributed to the simplification of matrix cracks simulation and compressive load at the interface in the thickness direction which is not taken into account.

The complicated process may cause the internal defects of multi-layered ceramic capacitors (MLCCs) and result in the malfunctions. This work aims to investigate the deformations of MLCCs that composed of nearly a hundred of BaTiO3 and Ni electrode films interleaved and stacked due to high pressure at elevated temperature. This study focuses on theoretical and numerical analyses. Classical laminated plate theory, linear elastic assumptions and equilibrium equations were adopted. Associated with the texts by Timoshenko and practical manufacturing process, three types of boundary conditions were considered, such as all edges simple-supported, two opposite edges simple-supported and the other two free, and four edges free. Also, two more conditions need be added, including four fixed points at corners and the elastic foundation at bottom. The numerical simulation by finite element method (FEM) incorporated with software ANSYS was used to obtain the displacement field of MLCCs due to high pressure at elevated temperature. The MLCCs were divided into nine regions with suitably different boundary conditions. Compared with the numerical results the analytical solutions of nine regions were found satisfactorily acceptable, i.e., the errors were about 0.1% - 6.2% for the boundary conditions of four edges free and four corners fixed. The errors about 0.13% - 6.15% were also acceptable for the boundary conditions of two opposite edges simple-supported and the others free. However, the analytical solutions did not agree with the numerical results for the case of all the boundary conditions simple-supported. Finally the proposed theoretical methodology provides an analytical method alternatively, instead of FEM and ANSYS, to analyze a nearly hundred layered MLCCs.

<p>The Swedish model for final disposal of nuclear fuel waste is based on copper canisters as a corrosion barrier with an inner pressure holding insert of cast iron. One of the methods to seal the copper canister is to use the Friction Stir Welding (FSW), a method invented by The Welding Institute (TWI).</p><p>This work has been focused on characterisation of the FSW joints, and modelling of the process, both analytically and numerically. The first simulations were based on Rosenthal’s analytical medium plate model. The model is simple to use, but has limitations. Finite element models were developed, initially with a two-dimensional geometry. Due to the requirements of describing both the heat flow and the tool movement, three-dimensional models were developed. These models take into account heat transfer, material flow, and continuum mechanics. The geometries of the models are based on the simulation experiments carried out at TWI and at Swedish Nuclear Fuel Waste and Management Co (SKB). Temperature distribution, material flow and their effects on the thermal expansion were predicted for a full-scale canister and lid. The steady state solutions have been compared with temperature measurements, showing good agreement.</p><p>Microstructure and hardness profiles have been investigated by optical microscope, Scanning Electron Microscope (SEM), Electron Back Scatter Diffraction (EBSD) and Rockwell hardness measurements. EBSD visualisation has been used to determine the grain size distribution and the appearance of twins and misorientation within grains. The orientation maps show a fine uniform equiaxed grain structure. The root of the weld exhibits the smallest grains and many annealing twins. This may be due to deformation after recrystallisation. The appearance of the nugget and the grain size depends on the position of the weld. A large difference can be seen both in hardness and grain size between the start of the weld and when the steady state is reached.</p>

Τα τροχαντήρια ενδομυελικά εμφυτεύματα τύπου Fi χρησιμοποιούνται σε περιπτώσεις μη σταθερών διατροχαντήριων καταγμάτων του ισχίου. Από κλινικής άποψης, δύο βασικά προβλήματα που αφορούν την αποτελεσματικότητα της χρήσης του ενδομυελικού εμφυτεύματος τύπου Fi είναι: ποια είναι η βέλτιστη θέση εισαγωγής της περιφερικής βίδας και το εάν πρέπει να χρησιμοποιηθεί μία περιφερική βίδα ή ένα ζεύγος περιφερικών βιδών. Προκειμένου να μελετηθούν οι επιδράσεις αυτών των δύο παραμέτρων, ένα σύνολο απλοποιημένων και πλήρως ολοκληρωμένων μοντέλων μοντέλων αναπτύχθηκαν και αναλύθηκαν με τη Μέθοδο των Πεπερασμένων Στοιχείων (ΜΠΣ). Η ανάλυση με χρήση της Μεθόδου των Πεπερασμένων Στοιχείων (ΜΠΣ) μπορεί να δώσει σημαντικές πληροφορίες που αφορούν στην ευαισθησία της συμπεριφοράς του εμφυτεύματος σε σχέση με τα άλλα τμήματα της κατασκευής του εμφυτεύματος (ήλος, lag βίδα, περιφερική βίδα). Σκοπός της εργασίας αυτής είναι η εύρεση της επίδρασης της θέσης της περιφερικής βίδας (των θέσεων των περιφερικών βιδών) στη μηχανική συμπεριφορά του συναρμολογήματος μηριαίου οστού/ενδομυελικού ήλου, όταν: 1.έχει ολοκληρωθεί η ενδομυελική οστεοσύνθεση 2.έχει διαγνωσθεί τυπικό διατροχαντήριο κάταγμα τύπου 31-Α1.3 κατά Müller AO/ASIF. / Trochanteric Fi-nail intramedullary fixation devices are used in cases of unstable intertrochanteric hip fractures. From a clinical point of view, two questions of high importance concerning the efficiency of the Fi-nail approach refer to where to put the distal screws and whether a single distal screw or a pair of distal screws should be used. In order to disclose the effect of those two parameters, a series of models were developed and then analyzed with the Finite Element Method (FEM). Finite element analysis can offer valuable information concerning the sensitivity of the implant’s behaviour in relation to the other parts of the implant’s structure (nail, lag screw, distal screw). The aim of this study is to examine the influence of the distal screw’s position in the mechanical behaviour of the femur/Fi-nail assembly: 1.when the intramedullary osteosynthesis has already been completed (case of healed femur) 2.when a typical intertrochanteric fracture (type 31-Α1.3, according to Müller AO/ASIF) has been diagnosed.

The aim of this thesis is the construction of a diagram of fracture strain for high strength steel OCHN3MFA and its application for the estimation of fracture toughness of this material. The first part of the thesis contains the necessary theoretical framework for numerical modelling of tensile tests of various specimens - smooth specimen, specimen with a notch and specimen with a circumferential crack, it describes the influence of triaxiality on the fracture of bodies and the possibilities of construction of the diagram of fracture strain. Subsequently, a numerical model of these specimens is created using the finite element method (FEM). A non-linear, elastoplastic calculation is performed. Fracture is identified by means of comparing the true stress versus the strain obtained from tests with the finite element analysis results. Stress–strain states of specimens at the moment of fracture are obtained from a numerical model. A diagram of fracture strain is constructed, and it is used to estimate the fracture toughness of a cracked body. Finally, a local approach, which uses the diagram of fracture strain, and a classical approach of fracture mechanics, especially the stress intensity factor, are compared.