<p>Failure along rock discontinuities can result in economic losses as well as loss of life. It is essential to develop methods that monitor the response of these discontinuities to shear loading to enable prediction of failure. Laboratory experiments are performed to investigate geophysical techniques to monitor shear failure of a pre-existing discontinuity to detect signatures of impending failure. Previous studies have detected precursors to shear failure in the form of maxima of transmitted waves across a discontinuity under shear. However, those experiments focused on well-matched discontinuities. However, in nature, rock discontinuities are not always perfectly matched because the asperities may be weathered by chemical, physical or mechanical processes. Further, the specific shear mechanism of mismatched discontinuities is still poorly understood. In this thesis, the ability to detect seismic precursors to shear failure for various discontinuity conditions—well-matched (rough and saw-tooth), mismatched (rough), and nonplanar (discontinuity profile with a half-cycle sine wave (HCS))—was assessed. The investigation was carried out through a coupled geophysical and mechanical experimental program that integrated detailed laboratory observations at the micro- and meso-scales. Shear experiments on gypsum discontinuities were conducted to observe changes in compressional (P) and shear (S) waves transmitted across the discontinuity. Digital Image Correlation (DIC) was used to quantify the vertical and horizontal displacements along the discontinuity during shearing to relate the location and magnitude of slip with the measured wave amplitudes. </p> <p>Results from the experiments conducted on planar, well-matched rough discontinuities (grit 36 sandpaper roughness) showed that seismic precursors to failure took the form of peaks in the normalized transmitted amplitude prior to the peak shear stress. Seismic wave transmission detected non-uniform dilation and closure of the discontinuity at a normal stress of 1 MPa. The results showed that large-scale roughness (presence of a HCS) could mask the generation of precursors, as it can cause non-uniform closure/dilation along the fracture plane at low normal stress. </p> <p>The experiments on idealized saw-toothed gypsum discontinuities showed that seismic precursors to failure appeared as maxima in the transmitted wave amplitude and conversely as minima in the reflected amplitudes. Converted waves (S to P & P to S) were also detected, and their amplitudes reached a maximum prior to shear failure. DIC results showed that slip occurred first at the top of the specimen, where the load was applied, and then progressed along the joint as the shear stress increased. This process was consistent with the order of emergence of precursors, i.e., precursors were first recorded near the top and later at the center, and finally at the bottom of the specimen. </p> <p>Direct shear experiments conducted on specimens with a mismatched discontinuity did not show any precursors (in the transmitted amplitude) to failure at low normal stresses (2 MPa), while those precursors appeared at higher normal stresses (5 MPa). The interplay between wave transmission, the degree of mismatch, and the discontinuity’s micro-physical, -chemical and -mechanical properties was assessed through: (1) 3D CT in-situ Xray scans to quantify the degree of mismatch at various normal stresses; (2) micro-indentation testing, to measure the micro-strength of the asperities; and (3) Scanning Electron Microscopy (SEM) and Electron Xray Diffraction (EDX), to study the micro-structure and chemical composition of the discontinuity. The X-ray results showed that contact between asperities increased with normal stress, even when the discontinuity was mismatched. The results indicated that: (1) at 2 MPa, the void aperture was large, so significant shear displacement was needed to interlock and damage the asperities; and (2) the micro-hardness of the asperities of the mismatched discontinuity was larger than that of the well-matched discontinuity, which points to inducing less damage for the same shear displacement. Both mechanisms contribute to the need for larger shear displacements to the mismatched discontinuity asperities to cause damage, which is consistent with the inability to detect seismic precursors to failure. The experimental results suggest that monitoring changes in transmitted wave amplitude across a discontinuity is a promising method for predicting impending failure for well-matched rock discontinuities. Precursor monitoring for mismatched rock discontinuities seems only possible when there is sufficient contact between the two rock surfaces, which occurs at large normal stresses. </p>

The short rate is central in the context of interest-rate markets as well as broader finance. As such, accurate modelling of this rate is of particular importance in the pricing of interest-rate options, especially during times of high volatility where increased demand is seen for simpler and lower risk investments. Recent interest has moved away from models of a pure continuous nature towards models that can account for discontinuities in the short rate. These are more representative of real world movements where the short rate is seen to jump due to current and scheduled market information. This dissertation examines this phenomenon in the context of a Vasicek short rate model and accounts for random-sized jumps at deterministic times following ideas similar to those introduced by Kim and Wright (2014). Finite difference methods are used successfully to find PDE solutions via backwards diffusion of the option value equation to its initial state. This procedure is implemented computationally and compared to Monte Carlo benchmark methods in order to assess its accuracy. In both non-jump and jump settings the method constructed was able to accurately price the call option specified and proved to be a viable means for pricing interest-rate options when stochastically-sized discontinuities are present at known times between inception and expiry. Furthermore the method showed that the stochastic discontinues in the short rate most notably affect the option price in the region around and just out of the money.

No description available.

Transverse zones are important syn-kinematic components of thrust belt development. Various scales of data were utilized to develop three-dimensional geometric and kinematic models for the Bessemer transverse zone (BTZ) of the Alabama Alleghanian thrust belt. Regional analysis of the BTZ began with the examination of geologic maps (1:250,000, 1:48,000, and 1:24,000 scales), seismic reflection profiles, well data, and previous stratigraphic research. All Paleozoic-age stratigraphic contacts, major thrust faults and associated folds, and various unnamed minor structures were compiled to create two strike-perpendicular, and five-strike parallel, cross sections transecting the extent of the BTZ at a scale of 1:100,000. The balanced and viable cross sections were used to create palinspastic maps of the BTZ. The deformed cross sections and geologic maps, and the restored cross sections and palinspastic maps, model the post- and prekinematic geometry of the transverse zone, respectively. Additional geological fieldwork in the northwestern part of the BTZ permitted the construction of geologic maps (1:24,000 scale) documenting cross-strike links (the fundamental unit of transverse zones) exposed at the present erosional surface (Concord and McCalla 7.5 quadrangles). Balanced and viable geologic cross sections (1:24,000 scale) were constructed from these data and placed parallel and perpendicular to strike of cross-strike links. The cross sections were restored and used to create 1:24,000-scale palinspastic maps of the cross-strike links in this part of the BTZ. The cross sections and maps model the three-dimensional geometry of the cross-strike links comprising the BTZ. Sub-allochthon basement structures are present beneath the thrust transport vectors of cross-strike links in the BTZ, indicating genetic relationships between transverse zone structures and underlying basment structures. Basement-graben related changes in the stratigraphic thickness of the decollement-host horizon are interpreted as having localized and facilitated growth of the Bessemer mushwad, a ductile duplex in the allochthon. The muswad localized the structural position of two thrust sheets and several cross-strike links in the BTZ. Geologic map patterns of the transverse zone indicate a break-back deformation sequence for the BTZ, interpreted as a response to decollement propagation through an allochthon-spanning weak decollement-host horizon, which had large stratigraphic thickness variations in basement grabens.

O comportamento geomecânico de túneis em rocha é fortemente influenciado pelas estruturas geológicas (descontinuidades) presentes no maciço. Do ponto de vista geométrico, estas estruturas são caracterizadas por parâmetros que descrevem suas orientações, frequências e comprimentos. Estes parâmetros, na maioria dos casos, são determinados em campanhas de campo, com bússola geológica e trena. Entretanto, limitações de tempo e acesso dificultam a execução destes trabalhos, impossibilitando a obtenção de números elevados de dados, que possibilitam análises estatísticas mais complexas. Para superar estas dificuldades, no presente estudo, o mapeamento das descontinuidades foi realizado em imagens do túnel Monte Seco, pertencente a Estrada de Ferro Vitória Minas (EFVM), obtidas por scanner a laser 3D. Neste mapeamento, tanto a orientação, quanto a posição e o comprimento dos traços das descontinuidades foram determinados com boa precisão, possibilitando a verificação da distribuição da intensidade de fraturamento de diferentes trechos, ao longo do túnel. Utilizando estes trechos diferenciados pela intensidade de fraturamento, foram elaboradas análises estatísticas mais complexas e adequadas (por janelas de amostragem e linhas de varredura) para determinação da orientação, comprimento médio dos traços e espaçamentos médios das famílias de descontinuidades. Com os parâmetros geométricos das descontinuidades, o modelo probabilístico de blocos rígidos foi construído, utilizando o software 3DEC. Assim, os parâmetros mecânicos das descontinuidades foram estimados utilizando correlações empíricas (a partir de descrições do maciço rochoso realizadas em mapeamentos geológicos por dentro do túnel), além de alguns ensaios de campo e laboratório. As análises com este modelo foram executadas para verificação da queda de blocos, e comparadas com as seções atuais do túnel. Os resultados indicaram que diferentes critérios de ruptura devem ser utilizados para diferentes tipos de descontinuidades (fraturas e foliação), e evidenciaram a importância de estimativas mais coerentes de parâmetros geométricos das descontinuidades nos resultados finais das análises. / The geomechanical behaviour of rock tunnels is strongly influenced by geological structures in the rock mass. Rock discontinuities are geometrically characterized by parameters that describes their orientations, frequency and lengths. In most cases, these parameters are determined in field inspections, using geological compass and measuring tapes. However, timeframes and access limitation hinder this procedure, making it impossible to obtain large amount of data that allow complex statistical analysis. To overcome these difficulties, here the discontinuity mapping was performed using images of the Monte Seco tunnel, obtained by 3D terrestrial laser scanning. In this case, the orientation, position and trace length of each discontinuity was determined with precision, allowing the verification of the fracture intensity distribution in different parts of the tunnel. Using these parts (differentiated by its fracture intensities), statistical analyses were performed, using sampling windows and scanlines, in order to determine the orientation mean trace length and spacing of discontinuity sets. Once the geometrical parameters of discontinuity sets were determined, a probabilistic model of rigid blocks was generated, using the 3DEC software. Thus, the mechanical parameters of discontinuity sets were estimated by empirical correlations (performed using descriptions of the rock mass obtained during geological inspections in the tunnel), and some laboratory and field tests. The analyses with this model were performed to verify the instability of blocks (block falls), and compared with actual cross sections of the tunnel. The results indicate that different failure criteria must be used for different discontinuity types (fractures and foliation), and revealed the importance of consistent estimated of geometrical parameters of discontinuity sets.

O comportamento geomecânico de túneis em rocha é fortemente influenciado pelas estruturas geológicas (descontinuidades) presentes no maciço. Do ponto de vista geométrico, estas estruturas são caracterizadas por parâmetros que descrevem suas orientações, frequências e comprimentos. Estes parâmetros, na maioria dos casos, são determinados em campanhas de campo, com bússola geológica e trena. Entretanto, limitações de tempo e acesso dificultam a execução destes trabalhos, impossibilitando a obtenção de números elevados de dados, que possibilitam análises estatísticas mais complexas. Para superar estas dificuldades, no presente estudo, o mapeamento das descontinuidades foi realizado em imagens do túnel Monte Seco, pertencente a Estrada de Ferro Vitória Minas (EFVM), obtidas por scanner a laser 3D. Neste mapeamento, tanto a orientação, quanto a posição e o comprimento dos traços das descontinuidades foram determinados com boa precisão, possibilitando a verificação da distribuição da intensidade de fraturamento de diferentes trechos, ao longo do túnel. Utilizando estes trechos diferenciados pela intensidade de fraturamento, foram elaboradas análises estatísticas mais complexas e adequadas (por janelas de amostragem e linhas de varredura) para determinação da orientação, comprimento médio dos traços e espaçamentos médios das famílias de descontinuidades. Com os parâmetros geométricos das descontinuidades, o modelo probabilístico de blocos rígidos foi construído, utilizando o software 3DEC. Assim, os parâmetros mecânicos das descontinuidades foram estimados utilizando correlações empíricas (a partir de descrições do maciço rochoso realizadas em mapeamentos geológicos por dentro do túnel), além de alguns ensaios de campo e laboratório. As análises com este modelo foram executadas para verificação da queda de blocos, e comparadas com as seções atuais do túnel. Os resultados indicaram que diferentes critérios de ruptura devem ser utilizados para diferentes tipos de descontinuidades (fraturas e foliação), e evidenciaram a importância de estimativas mais coerentes de parâmetros geométricos das descontinuidades nos resultados finais das análises. / The geomechanical behaviour of rock tunnels is strongly influenced by geological structures in the rock mass. Rock discontinuities are geometrically characterized by parameters that describes their orientations, frequency and lengths. In most cases, these parameters are determined in field inspections, using geological compass and measuring tapes. However, timeframes and access limitation hinder this procedure, making it impossible to obtain large amount of data that allow complex statistical analysis. To overcome these difficulties, here the discontinuity mapping was performed using images of the Monte Seco tunnel, obtained by 3D terrestrial laser scanning. In this case, the orientation, position and trace length of each discontinuity was determined with precision, allowing the verification of the fracture intensity distribution in different parts of the tunnel. Using these parts (differentiated by its fracture intensities), statistical analyses were performed, using sampling windows and scanlines, in order to determine the orientation mean trace length and spacing of discontinuity sets. Once the geometrical parameters of discontinuity sets were determined, a probabilistic model of rigid blocks was generated, using the 3DEC software. Thus, the mechanical parameters of discontinuity sets were estimated by empirical correlations (performed using descriptions of the rock mass obtained during geological inspections in the tunnel), and some laboratory and field tests. The analyses with this model were performed to verify the instability of blocks (block falls), and compared with actual cross sections of the tunnel. The results indicate that different failure criteria must be used for different discontinuity types (fractures and foliation), and revealed the importance of consistent estimated of geometrical parameters of discontinuity sets.

<p align="justify">In a context of restoration of historical masonry structures, it is crucial to properly estimate the residual strength and the potential structural failure modes in order to assess the safety of buildings. Due to its mesostructure and the quasi-brittle nature of its constituents, masonry presents preferential damage orientations, strongly localised failure modes and damage-induced anisotropy, which are complex to incorporate in structural computations. Furthermore, masonry structures are generally subjected to complex loading processes including both in-plane and out-of-plane loads which considerably influence the potential failure mechanisms. As a consequence, both the membrane and the flexural behaviours of masonry walls have to be taken into account for a proper estimation of the structural stability.</p> <p align="justify">Macrosopic models used in structural computations are based on phenomenological laws including a set of parameters which characterises the average behaviour of the material. These parameters need to be identified through experimental tests, which can become costly due to the complexity of the behaviour particularly when cracks appear. The existing macroscopic models are consequently restricted to particular assumptions. Other models based on a detailed mesoscopic description are used to estimate the strength of masonry and its behaviour with failure. This is motivated by the fact that the behaviour of each constituent is a priori easier to identify than the global structural response. These mesoscopic models can however rapidly become unaffordable in terms of computational cost for the case of large-scale three-dimensional structures.</p> <p align="justify">In order to keep the accuracy of the mesoscopic modelling with a more affordable computational effort for large-scale structures, a multi-scale framework using computational homogenisation is developed to extract the macroscopic constitutive material response from computations performed on a sample of the mesostructure, thereby allowing to bridge the gap between macroscopic and mesoscopic representations. Coarse graining methodologies for the failure of quasi-brittle heterogeneous materials have started to emerge for in-plane problems but remain largely unexplored for shell descriptions. The purpose of this study is to propose a new periodic homogenisation-based multi-scale approach for quasi-brittle thin shell failure.</p> <p align="justify">For the numerical treatment of damage localisation at the structural scale, an embedded strong discontinuity approach is used to represent the collective behaviour of fine-scale cracks using average cohesive zones including mixed cracking modes and presenting evolving orientation related to fine-scale damage evolutions.</p> <p align="justify">A first originality of this research work is the definition and analysis of a criterion based on the homogenisation of a fine-scale modelling to detect localisation in a shell description and determine its evolving orientation. Secondly, an enhanced continuous-discontinuous scale transition incorporating strong embedded discontinuities driven by the damaging mesostructure is proposed for the case of in-plane loaded structures. Finally, this continuous-discontinuous homogenisation scheme is extended to a shell description in order to model the localised behaviour of out-of-plane loaded structures. These multi-scale approaches for failure are applied on typical masonry wall tests and verified against three-dimensional full fine-scale computations in which all the bricks and the joints are discretised.</p>

This work presents a number of the practical aspects of developing reduced- order models (ROMs) based on proper orthogonal decomposition (POD). ROMS are derived and implemented for multiphase flow, quasi-2D nozzle flow and 2D inviscid channel flow. Results are presented verifying the ROMs against existing full-order models (FOM). POD is a method for separating snapshots of a flow field that varies in both time and space into spatial basis functions and time coefficients. The partial differential equations that govern fluid flow can then be pro jected onto these basis functions, generating a system of ordinary differential equations where the unknowns are the time coefficients. This results in the reduction of the number of equations to be solved from hundreds of thousands or more to hundreds or less. A ROM is implemented for three-dimensional and non-isothermal multiphase flows. The derivation of the ROM is presented. Results are compared against the FOM and show that the ROM agrees with the FOM. While implementing the ROM for multiphase flow, moving discontinuities were found to be a ma jor challenge when they appeared in the void fraction around gas bubbles. A point-mode POD approach is proposed and shown to have promise. A simple test case for moving discontinuities, the first order wave equation, is used to test an augmentation method for capturing the discontinuity exactly. This approach is shown to remove the unphysical oscillations that appear around the discontinuityin traditional approaches. A ROM for quasi-2D inviscid nozzle flow is constructed and the results are com- pared to a FOM. This ROM is used to test two approaches, POD-Analytical and POD-Discretized. The stability of each approach is assessed and the results are used in the implementation of a ROM for the Navier-Stokes equations. A ROM for a Navier-Stokes solver is derived and implemented using the results of the nozzle flow case. Results are compared to the FOM for channel flow with a bump. The computational speed-up of the ROM is discussed. Two studies are presented with practical aspects of the implementation of POD- based ROMs. The first shows the effect of the snapshot sampling on the accuracy of the POD basis functions. The second shows that for multiphase flow, the cross- coupling between field variables should not be included when computing the POD basis functions.

Today's product development process should be rapid andcost-efficient, and should result in innovative and reliableproducts. A crucial factor is the dynamic behaviour of theproduct. This thesis focuses on theoretical, numerical andexperimental approaches to achieve a comprehensiveunderstanding of dynamical phenomena occurring in nonlinearproducts, especially in products with parts that includeimpacts. The aim is to show the usefulness of nonlineartheories to better understand and optimise the dynamicbehaviour of products and thereby account for nonlinearphenomena already in the product development process. This is achieved through an investigation of researchefforts in the field of nonlinear dynamics; identification ofimportant research directions; a study on the effect ofcouplings between nonlinear parts; a detailed study on thedynamic behaviour of a product component; investigations oflow-cost strategies for controlling the dynamics of a nonlinearsystem; and the design and implementation of experimentalset-ups of two studied products. The investigation of research efforts shows that nonlinearparts are frequently included in products. Most common areparts that are nonlinear due to impacts and friction. Twoimportant areas are identified; to study coupling effectsbetween nonlinear subsystems and to study how nonlinearanalysis can be used to improve existing designs. Considering the studied products; a pantograph on a trainand a Braille printer, it can be concluded that thecharacteristics of a part can largely affect the dynamicbehaviour of the product. Typical nonlinear behaviour, such ascoexisting solutions and irregular motions, do occur. Theanalysis of the pan- tograph motion shows important aspects toconsider in the modelling process; coupling effects. In thecase of the Braille printer it is shown possible to create alow-cost control, by taking advantage of an existingdiscontinuity, to achieve a desired motion. Altogether, this work contributes to improved understandingof the be- haviour of nonlinear parts in products, especiallythose including impacts, pro- viding greater knowledge aboutaspects to consider in the design process. Keywords:Nonlinear Dynamics, Impacts, Discontinuities,Subsystems, Chaos, Irregular Behaviour, Printer Dynamics,Suspensions, Coupled Systems, Control. / QC 20100621

<p>For a successful design of an engineering system it is essential to pay careful attention to its dynamic response. This is particularly true, in the case of nonlinear systems, since they can exhibit very complex dynamic behaviour, including multiple co-existing stable solutions and chaotic motions, characterized by large sensitivity to initial conditions. In some systems nonlinear characteristics are desired and designed for, but in other cases they are unwanted and can cause fatigue and failure. A type of dynamical system which is highly nonlinear is discontinuous or non-smooth systems. In this work, systems with impacts are primarily investigated, and this is a typical example of a discontinuous system. To enhance or optimize the performance of dynamical systems, some kind of control can be implemented. This thesis concerns implementation of low-cost control strategies for discontinuous systems. Low-cost control means that a minimum amount of energy is used when performing the control actions, which is a desirable situation regardless of the application. The disadvantage of such a method is that the performance might be limited as compared with a control strategy with no restrictions on energy consumption. In this work, the control objective is to enforce a continuous or discontinuous grazing bifurcation of the system, whichever is desirable. In Paper A, the dynamic response and bifurcation behaviour of an impactoscillator with dry friction is investigated. For a one-degree-of-freedom model of the system, analytical solutions are found in separate regions of state space. These are then used to perform a perturbation analysis around a grazing trajectory. Through the analysis, a condition on the parameters of the system is derived, which assures a continuous grazing bifurcation. It is also shown that the result has bearing on the dynamic response of a two-degree-of-freedom model of the system. A low-cost active control strategy for a class of impact oscillators is proposed in Paper B. The idea of the control method is to introduce small adjustments in the position of the impact surface, at discrete moments in time, to assure a continuous bifurcation. A proof is given for what control parameters assures the stabilization. In Paper C, the proposed low-cost control method is implemented in a quarter-car model of a vehicle suspension, in order to minimize impact velocities with the bumpstop in case of high amplitude excitation. It is shown that the control method is effective for harmonic road excitation.</p>