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The singular solutions for linear elastostatics at corners are essential in both theory and computation. In this thesis, we seek new singular solutions for corners with the fixed (displacement), the free stress (traction) boundary conditions, and their mixed types, and to explore their corner singularity and provide the algorithms and error estimates in detail. The singular solutions of linear elastostatics are derived, and a number of new models of corner and crack singularity are proposed. Effective numerical methods, such as the collocation Trefftz methods (CTM), the method of fundamental solutions (MFS), the method of particular solutions (MPS) and their combinations: the so called combined method, are developed. Such solutions are useful to examine other numerical methods for singularity problems in linear elastostatics. This thesis consists of three parts, Part I: Basic approaches, Part II: Advanced topics, and Part III: Mixed types of displacement and traction conditions. Contents of Parts I and II have been published in [47,82]. In Part I, the collocation Trefftz methods are used to obtain highly accurate solutions, where the leading coefficient has 14 (or 13) significant digits by the computation with double precision. In part II, two more new models (symmetric and anti-symmetric) of interior crack singularities are proposed, for the corner and crack singularity problems, the combined methods by using many fundamental solutions, but by adding a few singular solutions are proposed. Such a kind of combined methods is significant for linear elastostatics with corners (i.e., the L-shaped domain), because the singular solutions can only be obtained by seeking the power £hk of r£hk numerically. Hence, only a few singular solutions used may greatly simplify the numerical algorithms; Part III is a continued study of Parts I and II, to explore mixed type of displacement and free traction boundary conditions. To our best knowledge, this is the first time to provide the particular solutions near the corner with mixed types of boundary conditions and to report their numerical computation with different boundary conditions on the same corner edge in linear elastostatics. This thesis explores corner singularity and its numerical methods, to form a systematic study of basic theory and advanced computation for linear elastostatics.

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Teeth trauma has become one of the most serious physical problems that people are suffering from in the past years. Early diagnosis and management are desperately needed to improve tooth survival, functionality and avoid the tooth loss. However, the diagnosis of cracked tooth could be challenging due to the small size of the crack. Therefore, a mechanics understanding of the tooth fracture is demanding from the perspective of developing a framework for failure prediction in clinical research and bio-mimetic restorative materials. This study focuses on characterising the anisotropic fracture behaviour and the crack shielding mechanisms in elephant dentin. This is often used as a structural analogue for human dentin due to the similarities in microstructure and chemical composition, in order to avoid the test-piece size restrictions, given the larger size of ivory than human teeth. Compact tension test-pieces were extracted from different locations on the ivory tusk so as to have different crack growth directions relative to the microstructure to inspect the fracture anisotropies. The fracture toughness as a function of the crack extension was assessed in terms of fracture resistance curves (R-curve). The accumulative crack-tip strain fields were also measured for the first time in dentin using digital image correlation technique (DIC) to investigate the capability of crack-tip elastic/plastic deformation before material failure. Investigation of crack morphologies, the interaction between crack and the microstructures, the fracture surfaces using both 2-D and 3-D techniques could provide with insights into extrinsic shielding mechanisms. Surface and volume crack opening displacement (COD) were measured for the first time optically and by X-ray computed tomography to investigate the effect of extrinsic crack-tip shielding. The displacement fields around the crack-tip obtained by DIC were fitted using Westergaard’s analytical solution to extract the effective stress intensity factor, by comparing this to the applied load, the efficiency of the crack-tip shielding could be evaluated. A novel cohesive element model (traction-separation law) was then established based on the COD results to simulate the physical process of crack-tip shielding. It is the first time the cohesive model has been adapted to studying the direct crack behaviour measured by in-situ experiment to predict the crack growth. This model was then validated using the crack-tip strain field and R-curve obtained from the experiment measurement.

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Les outillages de mise en forme à chaud sont soumis à des sollicitations thermomécaniques transitoires. Ils sont donc confrontés à la fissuration par fatigue. Le but de ce travail est d'étudier le comportement et la durée de vie en fatigue des aciers d'outillage, notamment leurs résistances à la propagation de fissure. Dans un premier temps, les chargements thermiques imposés aux outillages de fonderie sous pression de magnésium et d'emboutissage à chaud ont été évalués. Ensuite, le comportement et la durée de vie de l'acier à haute conductivité thermique HDC1 ont été étudiés et comparés à l'acier AISI H11 (acier de référence) en fonction de la température. L'acier HDC1 présente un adoucissement cyclique stable à 20°C et 300°C. Par contre, l'intensité d'adoucissement est plus importante à hautes températures. La durée de vie a été étudiée en utilisant les lois de Manson-Coffin et de Basquin. A hautes températures, l'oxydation devient un mécanisme d'endommagement primordial pour l'acier HDC1 et provoque des durées de vie plus courtes que celles observées sur l'acier AISI H11. La résistance à la propagation de fissure de fatigue a été déterminée dans des aciers à la température ambiante par le biais de critères : l'amplitude de facteur d'intensité de contrainte (∆K) et l'amplitude d'ouverture en pointe de fissure (∆CTOD). La méthode de corrélation d'images a permis de mesurer (∆COD) et d'évaluer (∆CTOD). L'ensemble de ces résultats a permis de mettre en évidence l'effet de fermeture de fissure et le comportement plastique en pointe de fissure. ∆CTOD présente un bon critère pour rationaliser la propagation de fissure dans les aciers étudiés.Enfin, la simulation numérique de la propagation de fissure en fatigue a été menée dans l'acier AISI H11 à 600°C par la méthode de relâchement des nœuds en éléments finis. L'effet des modèles de comportement monotone (élastoplastique) et cyclique (élastoviscoplastique) a été étudié sur le calcul de l'ouverture de fissure et la plasticité en pointe de fissure. L'influence du modèle de comportement est faible sur le calcul de l'amplitude d'ouverture de fissure ∆COD, du fait d'une plasticité confinée en pointe de fissure. Alors que, le modèle de comportement cyclique est mieux adapté pour décrire la plasticité en pointe de fissure. / Hot metal forming tools are subjected to cyclic thermomechanical loading and damage by complex fatigue/wear/oxidation interactions. Thermal solicitations were measured on high pressure die casting and hot stamping processes. Based on thermal measurements, the isothermal fatigue behaviour and lifetime of a new high conductivity steel HDC1 were investigated at different temperatures and strain amplitude then compared to AISI H11 steel. As AISI H11, continue cyclic softening was observed in HDC1 at all temperatures. The Manson-Coffin and Basquin laws were used for life prediction models under different temperatures. It was observed that the fatigue/oxidation interaction was a principal damage mechanism of the HDC1 steel at high temperature. Fatigue crack propagation in steels was investigated at room temperature in SENT specimens. A digital image correlation technique was used to evaluate crack opening (∆COD) and crack tip opening displacement (∆CTOD) ranges. Crack growth rate were investigated using ∆K (Paris law) and ∆CTOD criteria. It was observed that the cyclic crack tip plasticity control the crack propagation resistance. Crack closure could be evaluated by ∆CTOD.Finite element method by debond technique was used to model the crack propagation of AISI H11 at 600°C using both monotonic elasto-plastic (EP) and cyclic elasto-viscoplastic (EVP) constitutive laws materials. The comparison of ∆COD calculated and measured had shown that monotonic EP and cyclic EVP had no significant effect on the ∆COD, mainly due to the small-scale yielding conditions. It is however observed that the cyclic constitutive law was the best suitable model for the crack tip plasticity effect.

De nombreux phénomènes à l’oeuvre dans le processus de déformation et d’endommagement des élastomères sont étudiés à partir de la réponse mécanique de ces matériaux. Cependant, la plupart de ces phénomènes dépendent de la température et ont des signatures calorimétriques qui pourraient permettre de mieux les comprendre. Dans le contexte industriel de la manufacture de pneumatiques, les élévations de température induites par le chargement peuvent fortement impacter les performances physiques des pneumatiques ainsi que la tenue en fatigue des constituants caoutchoutiques. L’objectif de cette thèse est donc de caractériser le comportement thermomécanique des élastomères chargés et non chargés sous divers types de chargement mécanique. Pour ce faire, des mesures de champs thermiques et cinématiques couplées ont été mises en oeuvre lors d’essais mécaniques à température ambiante. Dans un premier temps, des essais de traction uniaxiale sur éprouvettes indemnes ont permis de confirmer que l’hystérésis mécanique observée lors d’un cycle de traction sur un caoutchouc naturel non chargé est essentiellement due au phénomène de cristallisation et non à des phénomènes dissipatifs. Par ailleurs, la construction de bilans énergétiques sur un cycle mécanique a permis de distinguer la contribution des différents mécanismes dissipatifs (viscosité, effet Mullins) des couplages thermomécaniques (élasticité entropique, cristallisation). Dans un second temps, des essais de cisaillement pur ont été menés sur des éprouvettes préalablement entaillées. Les analyses thermomécaniques menées à l’échelle de la zone d’influence de la fissure ont montré que les phénomènes dissipatifs aux très grandes déformations ne s’expriment pas de la même manière qu’aux déformations plus faibles. En particulier, pour les mélanges considérés dans cette étude, les effets du couplage entropique et de la viscosité sont du même ordre à la décharge, si bien que le matériau n’absorbe pas de chaleur à la décharge. Ces résultats sont très prometteurs à la fois pour la compréhension des phénomènes physiques impliqués dans le processus de déformation et pour la modélisation du comportement thermomécanique des élastomères. / Usually, most of the physical phenomena involved in the deformation of elastomers are studied from purely mechanical approaches. However, almost all of such phenomena depend on temperature and have distinguishable calorimetric signatures, which can enable us to better understand them. Furthermore, in the tire industrial context, the temperature increase induced by loading and self-heating may strongly impact the physical performances of tires as well as the fatigue life of the rubber components. Consequently, the aim of this PHD thesis was to characterize the thermomechanical behavior of rubbers, filled and unfilled, subjected to different mechanical loadings. For this purpose, coupled thermal and kinetic full field measurements have been performed during mechanical tests at ambient temperature. First, homogeneous uniaxial tensile tests have enabled us to confirm that the hysteresis loop in terms of the stress-strain relationship is mainly induced by crystallization phenomenon in natural rubber, not to dissipative phenomena. In the same way, energetic balances over one mechanical cycle have enabled us to distinguish the contribution of different dissipative phenomena (viscosity, Mullins effect) and the thermomechanical couplings (entropic elasticity, crystallization). Second, the analysis of the pure shear tests with pre-cracked specimens has enabled us to analyze, for the first time, the calorimetric response of rubbers in the zone of crack influence. Results have shown that dissipative phenomena at large strains differ from those involved at smaller strains. More especially, for the materials considered in the present study, the effects of the entropic coupling and viscosity are of the same order of magnitude during unloading, so that the material does not absorb any heat during unloading. These results are promising and motivate further work in this field in order to better understand the physical phenomena involved in the deformation processes as well as to more relevantly model the thermomechanical behavior of elastomers.

In order to understand and predict the fracture behaviour of polycrystalline materials from a fundamental perspective, it is important to first investigate plastic deformation at a crack tip in a ductile single crystal. In this context, it may be noted that when the crack opening displacement is much less than the grain size, the crack tip fields are entirely contained in a single grain. Further, some key structural components are being fabricated in single crystal form. For example, blades in high pressure turbines of jet engines are made of single crystals of Nickel-based superalloys. In view of the above considerations, a combined experimental and computational study of the crack tip stress and strain fields in FCC single crystal is carried out in the present work. The effect of constraint level, which is characterized by the T-stress under mode I, plane strain small scale yielding conditions, on the near-tip response is first analyzed for a crystal orientation in which the crack plane coincides with (010) and ¯the crack front lies along[101]direction. A family of finite element solutions are generated by employing a boundary layer approach within continuum crystal plasticity framework. The results show that the near-tip deformation field, especially the development of kink and slip shear bands, is sensitive to the constraint level. On imposition of negative T-stress, a significant drop in the hydrostatic stress level is noticed in the region ahead of the tip. This suggests loss of crack tip constraint with negative T-stress, which is akin to isotropic plastic solids. The reason for the loss of crack tip constraint is traced to the occurrence of an elastic sector near the notch tip. The results also show that a two-parameter (such as K-T or J-Q) characterization of near-tip fields is necessary to accommodate different constraint levels in FCC single crystals. The results of the boundary layer formulation are used to guide the construction of asymptotic solutions near the crack tip corresponding to various constraint levels in elastic-perfectly plastic FCC single crystal. Two families of alternate asymptotic solutions are constructed by introducing an elastic near-tip sector. These families of stress fields are parameterized by the normalized opening stress ahead of the tip, τA22/τo, where τo is the critical resolved shear stress, and a quantity (p) which characterizes the coordinates of the point where elastic unloading commences in stress plane. The results show that the stress distribution corresponding to each member of these families, as well as the trajectories in stress plane as the crack tip is traversed, agree well with finite element results for a certain value of T-stress. In order to validate the above numerical and analytical solutions, the nature of crack tip deformation in aluminium single crystals is examined experimentally in a high constraint three point bend (TPB) specimen and in a low constraint single edge notch tensile (SENT) geometry. These experiments provide evidence, based on in-situ Electron Back Scattered Diffraction (EBSD) of the existence of kink shear bands (involving lattice rotation) exactly as predicted by Rice [J.R. Rice, Mech. Mater. 6 (1987) 317] and the present finite element analysis. The experimental investigation of a low constraint SENT geometry is also supplemented by 3D finite element computations based on continuum crystal plasticity. These computational results enable assessment of 3D effects near the tip. Finally, the effects of different lattice orientations (especially ones for which the slip systems are not symmetric with respect to the notch line) on the near-tip fields are studied pertaining to various constraint levels. The results obtained for different orientations show that the near-tip deformation field is sensitive to the constraint level. The stress distribution and the size and shape of plastic zone near the notch tip are also strongly influenced by the level of T-stress. It is clearly established that ductile single crystal fracture geometries, would progressively lose stress triaxiality with increase in negative T-stress irrespective of lattice orientation. Also, the near-tip field is shown to be part of a family which can be characterized by two parameters (such as K – T or J - Q).

Double cantilever beam (DCB) specimens with distinct intermetallic microstructures and different geometries were fractured under different mode ratios of loading, ψ, to obtain critical strain energy release rate, Jc. The strain energy release rate at crack initiation, Jci, increased with phase angle, ψ, but remained unaffected by the joint geometry. However, the steady-state energy release rate, Jcs, increased with the solder layer thickness. Also, both the Jci and Jcs decreased with the thickness of the intermetallic compound layer. Next, mode I and mixed-mode fracture tests were performed on discrete (l=2 mm and l=5 mm) solder joints arranged in a linear array between two copper bars to evaluate the J = Jci (ψ) failure criteria using finite element analysis. Failure loads of both the discrete joints and the joints in commercial electronic assemblies were predicted reasonably well using the Jci from the continuous DCBs. In addition, the mode-I fracture of the discrete joints was simulated with a cohesive zone model which predicted reasonably well not only the fracture loads but also the overall load-displacement behavior of the specimen. Additionally, the Jci calculated from FEA were verified estimated from measured crack opening displacements in both the continuous and discrete joints. Finally, the pad-crater fracture mode of solder joints was characterized in terms of the Jci measured at various mode ratios, ψ. Specimens were prepared from lead-free chip scale package-PCB assemblies and fractured at low and high loading rates in various bending configurations to generate a range of mode ratios. The specimens tested at low loading rates all failed by pad cratering, while the ones tested at higher loading rates fractured in the brittle intermetallic layer of the solder. The Jci of pad cratering increased with the phase angle, ψ, but was independent of surface finish and reflow profile. The generality of the J =Jci(ψ) failure criterion to predict pad cratering fracture was then demonstrated by predicting the fracture loads of single lap-shear specimens made from the same assemblies.

Double cantilever beam (DCB) specimens with distinct intermetallic microstructures and different geometries were fractured under different mode ratios of loading, ψ, to obtain critical strain energy release rate, Jc. The strain energy release rate at crack initiation, Jci, increased with phase angle, ψ, but remained unaffected by the joint geometry. However, the steady-state energy release rate, Jcs, increased with the solder layer thickness. Also, both the Jci and Jcs decreased with the thickness of the intermetallic compound layer. Next, mode I and mixed-mode fracture tests were performed on discrete (l=2 mm and l=5 mm) solder joints arranged in a linear array between two copper bars to evaluate the J = Jci (ψ) failure criteria using finite element analysis. Failure loads of both the discrete joints and the joints in commercial electronic assemblies were predicted reasonably well using the Jci from the continuous DCBs. In addition, the mode-I fracture of the discrete joints was simulated with a cohesive zone model which predicted reasonably well not only the fracture loads but also the overall load-displacement behavior of the specimen. Additionally, the Jci calculated from FEA were verified estimated from measured crack opening displacements in both the continuous and discrete joints. Finally, the pad-crater fracture mode of solder joints was characterized in terms of the Jci measured at various mode ratios, ψ. Specimens were prepared from lead-free chip scale package-PCB assemblies and fractured at low and high loading rates in various bending configurations to generate a range of mode ratios. The specimens tested at low loading rates all failed by pad cratering, while the ones tested at higher loading rates fractured in the brittle intermetallic layer of the solder. The Jci of pad cratering increased with the phase angle, ψ, but was independent of surface finish and reflow profile. The generality of the J =Jci(ψ) failure criterion to predict pad cratering fracture was then demonstrated by predicting the fracture loads of single lap-shear specimens made from the same assemblies.