Endodontic retreatments are time-consuming and difficult to perform. Removing the root canal filling material of the original treatment with maximum efficiency and safety is a complex task that requires a good technique, patience and perseverance. This filling material is often contaminated and prevents access to areas of the canal harboring bacteria and remnants of necrotic tissue, especially in the isthmus region and in dentinal tubules. The aim of this study was to use micro-computed tomography to evaluate two protocols for the removal of root canal filling material from curved root canals of the mesial roots of mandibular molars using a reciprocating instrument followed by two different rotary instruments manufactured with different nickel-titanium alloys. The reduction in the volume of the filling material inside the root canals was evaluated. Assessments were also made of the influence of the glide path extension on root canal obturation, the extrusion of filling material after the use of the reciprocating instrument, the volume of filling material after retreatment and its presence in isthmus areas. In addition, the use of a solvent and the agitation of a 2% sodium hypochlorite solution using either passive ultrasonic irrigation (PUI) or a plastic instrument in reciprocating motion were evaluated. A total of 40 mesial roots of mandibular molars with a mean curvature of 25 degrees were selected. The teeth were radiographed digitally in the buccolingual and mesiodistal directions, with a digital Xray system (Schick CDR; Schick Technologies, Long Island, NY), using an exposure time of 0.16 s. This was done in order to select the teeth and then compose two groups with similar characteristics to ensure their anatomic homogeneity. Group homogeneity in terms of anatomy, root canal length, curvature angle, and Vertucci classification type II and IV were confirmed and showed that the study groups were well balanced. Two groups were defined according to the retreatment protocol adopted, namely RHPui (Reciproc + Hyflex + PUI) and RMEasy (Reciproc + Mtwo + EasyClean). The teeth were instrumented with the Reciproc R25 instrument and filled using the single-cone technique and AHPlus cement. Prior to the use of this instrument, canal patency was achieved by creating a glide path with PathFile instruments, throughout the entire extension of the canal in Group RHPui, and up to 1 mm short of the total canal length in Group RMEasy. In both groups, the original filling material was removed with the Reciproc R25 instrument. Xylene was used as a solvent in Group RHPui up to the middle third, but not in the apical area, in the later steps of instrumentation with Reciproc R25. In Group RHPui, the Hyflex 40.04 instrument, made with a CM nickeltitanium alloy, was used. After that, xylene solvent was applied for one minute in the pulp chamber, followed by passive ultrasonic irrigation (PUI) with 2% sodium hypochlorite using an irrisonic instrument, applied up to 2 mm short of the working length. In Group RMEasy, the Mtwo 40.04 instrument, made with a conventional nickeltitanium alloy, was used. Xylene solvent was also used in the pulp chamber for one minute, followed by application of a 2% sodium hypochlorite solution agitated mechanically with an EasyClean plastic instrument in reciprocating motion up to the full working length. The two groups were re-obturated using the warm vertical condensation technique, with a gutta-percha cone corresponding to the final instrument, and with AHPus cement. Micro-CT scans were performed after each procedure to evaluate the volume of the remaining material, the extrusion after the use of the reciprocating instruments, the final volume of the obturation at the end of the retreatment and the filling material volume in the isthmus area, both after the original obturation and after re-obturation. The results showed that the initial filling was significantly closer to the apical foramen in Group RHPui than in Group RMEasy, after the initial treatment. After performing the retreatment protocols, the complete removal of filling material from inside the root canals was not achieved in either study group, with no statistically significant difference (P > 0.05) between groups regarding this variable. In the intra-group evaluation, a statistically significant difference (P < 0.05) was found between the procedural steps regarding the reduction in filling material volume, both in the total canal length and at its different levels, except for the middle third after HyFlex, the cervical third after Mtwo, and the middle third after EasyClean. The volume of the final filling material was significantly higher (P < 0.05) than the volume of the original filling material in both study groups. The volume of filling material in the isthmus region was significantly higher after re-obturation in both groups (P < 0.05), and this increase was greater in the RMEasy Group. There was no statistically significant difference (P > 0.05) between groups regarding the total amount of time expended in the removal of filling material (P > 0.05), namely 209.40 s for the RHPui Group and 227.40 s for the RMEasy Group. There was also no statistically significant difference (P > 0.05) between groups regarding the volume of extruded filling material, namely 0.0258 mm3 (0 0.954) in the RHPui Group and 0.0037 mm3 (0 0.565) in the RMEasy Group. The Shapiro-Wilks, Mann-Whitney, Wilcoxon, Friedman and Dunn tests were used in the data analysis. / Os retratamentos endodônticos demandam tempo e são difíceis de serem executados. Remover o material de obturação do tratamento original com a máxima eficiência e segurança é uma tarefa complexa que necessita de técnica, paciência e perseverança. Esse material de obturação muitas vezes está contaminado, e impede o acesso às áreas do canal onde permaneceram restos de tecido necrótico e microrganismos, especialmente na região do istmo e nos túbulos dentinários. O objetivo deste trabalho foi avaliar, por meio da microtomografia computadorizada, dois protocolos de desobturação de canais radiculares curvos de raízes mesiais de molares inferiores utilizando um instrumento reciprocante seguido de dois diferentes instrumentos rotatórios fabricados com diferentes ligas metálicas de níquel-titânio. Além da redução dos volumes de material obturador no interior dos canais, foram avaliados a influência do comprimento da patência do canal radicular (glide path) na obturação, extrusão de material obturador após o uso do instrumento reciprocante Reciproc R25, o volume do material de obturação após o retratamento e a presença deste em áreas de istmo. Também foi avaliado o uso de um solvente seguido da agitação do hipoclorito de sódio a 2% com ultrassom (PUI) e com um instrumento plástico acionado em movimento reciprocante. Foram selecionadas 40 raízes mesiais de molares inferiores com curvatura média de 25 graus que foram divididos em dois grupos de forma homogênea quanto ao comprimento, à curvatura e à anatomia (classes II e IV de acordo com a classificação de Vertucci) dos canais radiculares. Os dentes foram instrumentados com o instrumento Reciproc R25 e obturados com a técnica do cone único e com cimento AHPlus. Previamente ao uso do instrumento Reciproc R25, obteve-se a patência inicial do canal, ou seja, criou-se um glide path, com os instrumentos PathFile em toda a extensão do canal no Grupo RHPui e 1 mm aquém do comprimento total do canal no Grupo RMEasy. Os dois grupos foram definidos de acordo com o protocolo de retratamento adotado, a saber, RHPui (Reciproc + Hyflex + PUI) e RMEasy (Reciproc + Mtwo + EasyClean). Nos dois grupos, a remoção do material obturador original foi realizada com o instrumento Reciproc R25, sendo que no grupo RHPui foi utilizado o solvente xilol até a proximidade do terço médio, não sendo utilizado solvente no terço apical. No Grupo RHPui, utilizouse, então, o instrumento Hyflex 40.04 feito com liga de niquel-titânio CM (control memory ou de memória controlada) e, por último, utilizou-se o solvente xilol por um minuto na câmara pulpar, seguido de irrigação ultrassônica passiva (PUI) com solução de hipoclorito de sódio a 2% utilizando o instrumento Irrisonic, até o limite de 2 mm aquém do comprimento de trabalho. No Grupo RMEasy, foi utilizado o instrumento Mtwo 40.04 fabricado com liga de níquel-titânio convencional e, a seguir, também se utilizou o solvente xilol por um minuto na câmara pulpar seguido de uma agitação mecânica do hipoclorito de sódio a 2% com o instrumento plástico EasyClean em movimento reciprocante até o limite do comprimento total de trabalho. Os dois grupos foram re-obturados com a técnica de condensação vertical aquecida com cone de guta-percha correspondente ao instrumento final e com cimento AHPus. Foram realizadas microtomografias após cada procedimento para avaliar o volume de material remanescente, a extrusão após o uso dos instrumentos reciprocantes, o volume final da obturação ao final do retratamento e o volume na área de istmo, tanto após a obturação original quanto após a re-obturação. Os resultados obtidos mostraram que, no tratamento inicial, a patência do canal (ou glidepath) realizado com instrumentos Pathfile previamente ao uso dos instrumentos Reciproc R25 possibilitou que a obturação dos canais atingisse um limite significativamente (P < 0,05) mais próximo ao forame apical no grupo RHPui, onde o glidepath foi realizado em toda a extensão do canal. Em relação aos diferentes protocolos de retratamento, não houve remoção completa do material obturador do interior dos canais nos dois grupos estudados em todos os níveis avaliados, sem diferença estatisticamente significativa (P > 0,05) entre os grupos com relação a essa variável. Na avaliação intragrupo, houve uma diferença estatisticamente significativa (P < 0,05) entre os passos do procedimento em relação à redução do volume de material obturador, tanto na extensão total do canal, quanto nos seus diferentes níveis, com exceção do terço médio após o Hyflex, no terço cervical após o Mtwo e no terço médio após o EasyClean. O volume do material obturador final foi significativamente maior (P < 0,05) do que o volume do material obturador original, nos dois grupos estudados. O volume de material obturador na região do istmo foi significativamente maior (P < 0,05) após a re-obturação nos dois grupos, sendo significativamente maior (P < 0,05) no grupo RMEasy do que no grupo RHPui. Não houve diferença estatística (P > 0,05) entre os grupos com relação ao tempo total despendido na remoção do material obturador (P > 0,05), sendo 209,40 s para o Grupo RHPui e 227,40 s para o Grupo RMEasy. Não houve diferença estatística (P > 0,05) entre os grupos com relação ao volume da extrusão de material obturador, sendo 0,0258 mm3 (0 0,954) no Grupo RHPui e 0,0037 mm3 (0 0,565) no Grupo RMEasy. Os testes de Shapiro-Wilks, Mann- Whitney, Wilcoxon, Friedman e Dunn foram utilizados na análise dos dados.

Po zhrnutí potrebných teoretických základov a predošlého výskumu v ob- lasti, prezentujeme semi-klasickú kvantovaciu schému pre uzavretú Nambuovu- Gotovu strunu. Týmto zovšeobecňujeme predošlú prácu, ktorá bola vykonaná pre otvorenú strunu a uzavretú strunu kruhového tvaru. Pomocou metód kvan- tovej teórie po©a v zakrivených priestoročasoch počítame strednú hodnotu vo©- ného Hamiltoniánu struny rotujúcej v dvoch na seba kolmých priestorových rovinách v priestoročase všeobecnej dimenzie. Táto hodnota je priamo úmerná kvantovej korekcii k celkovej energii struny, ktorá má formu tzv. Reggeovho interceptu. Výslednú hodnotu Reggeovho interceptu porovnávame s predošlým výskumom. Taktiež uvádzame porovnanie získaného spektra fyzikálnych stavov struny kvantovanej našou metódou so spektrom odvodeným pomocou kovariant- ného kvantovania. 1

The future International Linear Collider (ILC) and Compact Linear Collider (CLIC) are intended for precision measurements of phenomena discovered at the Large Hadron Collider (LHC) and also for the discovery of new physics. In order to offer optimal conditions for such experiments, the new colliders must produce very-high-luminosity collisions at energies in the TeV regime. Emittance growth caused by imperfections in the main linacs is one of the factors limiting the luminosity of CLIC and ILC. In this thesis, various emittance preservation and luminosity tuning techniques have been tested and developed in order to meet the challenging luminosity requirements. Beam-based alignment was shown to be insufficient for reduction of emittance growth. Emittance tuning bumps provide an additional powerful preservation tool. After initial studies of tuning bumps designed to treat certain imperfections, a general strategy for design of optimised bumps was developed. The new bumps are optimal both in terms of emittance reduction performance and convergence speed. They were clearly faster than previous bumps and reduced emittance growth by nearly two orders of magnitude both for CLIC and ILC. Time-dependent imperfections such as ground motion and magnet vibrations also limit the performance of the colliders. This type of imperfections was studied in detail, and a new feedback system for optimal reduction of emittance growth was developed and shown to be approximately ten times more efficient than standard trajectory feedbacks. The emittance tuning bumps require fast and accurate diagnostics. The possibility of measuring emittance using a wide laserwire was introduced and simulated with promising results. While luminosity cannot be directly measured fast enough, it was shown that a beamstrahlung tuning signal could be used for efficient optimisation of a number of collision parameters using tuning bumps in the Final Focus System. Complete simulations of CLIC emittance tuning bumps, including static and dynamic imperfections and realistic tuning and emittance measurement procedures, showed that an emittance growth six times lower than that required may be obtained using these methods.

The modified mapping-collocation (MMC) method is applied to analyze a circumferential crack in an isotropic curved beam. Based on the method a MATLAB code was developed to obtain the stress field. Incorporating the stress correlation technique, the opening and sliding fracture mode stress intensity factors (SIF)s of the crack for the beam under pure bending moment load case are calculated. The MMC method aims to solve two-dimensional problems of linear elastic fracture mechanics (LEFM) by combining the power of analytic tools of complex analysis (Muskhelishvili formulation, conformal mapping, and extension arguments) with simplicity of applying the boundary collocation method as a numerical solution approach. Qualitatively, a good agreement between the computed stress contours and the fringe shapes obtained from the photoelastic experiment on a plexiglass specimen is observed. Quantitatively, the results are compared with that of ANSYS finite element analysis software. The effect of crack size, crack position and beam thickness variation on SIF values and mode mixity is investigated.

This thesis investigates the strength behavior of a representative highly skewed and horizontally curved bridge as well as analysis and design procedures for these types of structures. The bridge responses at and above a number of limits in the AASHTO (2007) Specifications are considered. The study includes the evaluation of various attributes of the elastic analysis of the subject bridge. These attributes include: (1) the accuracy of 3-D grid versus 3-D FEA models, (2) first-order versus second-order effects during the construction, (3) the ability to predict layover at bearing lines using simplified equations and (4) the benefit of combining the maximum and concurrent major-axis and flange lateral bending values due to live load compared to combining the maximums due to different live loads when checking the section resistances. The study also addresses the ability of different AASHTO 2007 resistance equations to capture the ultimate strength behavior. This is accomplished by comparing the results from full nonlinear 3-D FEA studies to the elastic design and analysis results. Specifically the use of the 2007 AASHTO moment based one-third rule equations is evaluated for composite sections in positive bending.

Uncertainties in load and system properties play a significant role in reliability analysis of vibrating structural systems. The subject of random vibrations has evolved over the last few decades to deal with uncertainties in external loads. A well developed body of literature now exists which documents the status of this subject. Studies on the influ­ence of system property uncertainties on reliability of vibrating structures is, however, of more recent origin. Currently, the problem of dynamic response characterization of sys­tems with parameter uncertainties has emerged as a subject of intensive research. The motivation for this research activity arises from the need for a more accurate assess­ment of the safety of important and high cost structures like nuclear plant installations, satellites and long span bridges. The importance of the problem also lies in understand­ing phenomena like mode localization in nearly periodic structures and deviant system behaviour at high frequencies. It is now well established that these phenomena are strongly influenced by spatial imperfections in the vibrating systems. Design codes, as of now, are unable to systematically address the influence of scatter and uncertainties. Therefore, there is a need to develop robust design algorithms based on the probabilistic description of the uncertainties, leading to safer, better and less over-killed designs. Analysis of structures with parameter uncertainties is wrought with diffi­culties, which primarily arise because the response variables are nonlinearly related to the stochastic system parameters; this being true even when structures are idealized to display linear material and deformation characteristics. The problem is further com­pounded when nonlinear structural behaviour is included in the analysis. The analysis of systems with parameter uncertainties involves modeling of random fields for the system parameters, discretization of these random fields, solutions of stochastic differential and algebraic eigenvalue problems, inversion of random matrices and differential operators, and the characterization of random matrix products. It should be noted that the mathematical nature of many of these problems is substantially different from those which are encountered in the traditional random vibration analysis. The basic problem lies in obtaining the solution of partial differential equations with random coefficients which fluctuate in space. This has necessitated the development of methods and tools to deal with these newer class of problems. An example of this development is the generalization of the finite element methods of structural analysis to encompass problems of stochastic material and geometric characteristics. The present thesis contributes to the development of methods and tools to deal with structural uncertainties in the analysis of vibrating structures. This study is a part of an ongoing research program in the Department, which is aimed at gaining insights into the behaviour of randomly parametered dynamical systems and to evolve computational methods to assess the reliability of large scale engineering structures. Recent studies conducted in the department in this direction, have resulted in the for­mulation of the stochastic dynamic stiffness matrix for straight Euler-Bernoulli beam elements and these results have been used to investigate the transient and the harmonic steady state response of simple built-up structures. In the present study, these earlier formulations are extended to derive the stochastic dynamic stiffness matrix for a more general beam element, namely, the curved Timoshenko beam element. Furthermore, the method has also been extended to study the mean and variance of the stationary response of built-up structures when excited by stationary stochastic forces. This thesis is organized into five chapters and four appendices. The first chapter mainly contains a review of the developments in stochas­tic finite element method (SFEM). Also presented is a brief overview of the dynamics of curved beams and the essence of the dynamic stiffness matrix method. This discussion also covers issues pertaining to modeling rotary inertia and shear deformations in the study of curved beam dynamics. In the context of SFEM, suitability of different methods for modeling system uncertainties, depending on the type of problem, is discussed. The relative merits of several schemes of discretizing random fields, namely, local averaging, series expansions using orthogonal functions, weighted integral approach and the use of system Green functions, are highlighted. Many of the discretization schemes reported in the literature have been developed in the context of static problems. The advantages of using the dynamic stiffness matrix approach in conjunction with discretization schemes based on frequency dependent shape functions, are discussed. The review identifies the dynamic analysis of structures built-up of randomly parametered curved beams, using dynamic stiffness matrix method, as a problem requiring further research. The review also highlights the need for studies on the treatment of non-Gaussian nature of system parameters within the framework of stochastic finite element analysis and simulation methods. The problem of deterministic analysis of curved beam elements is consid­ered first. Chapter 2 reports on the development of the dynamic stiffness matrix for a curved Timoshenko beam element. It is shown that when the beam is uniformly param-etered, the governing field equations can be solved in a closed form. These closed form solutions serve as the basis for the formulation of damping and frequency dependent shape functions which are subsequently employed in the thesis to develop the dynamic stiffness matrix of stochastically inhomogeneous, curved beams. On the other hand, when the beam properties vary spatially, the governing equations have spatially varying coefficients which discount the possibility of closed form solutions. A numerical scheme to deal with this problem is proposed. This consists of converting the governing set of boundary value problems into a larger class of equivalent initial value problems. This set of Initial value problems can be solved using numerical schemes to arrive at the element dynamic stiffness matrix. This algorithm forms the basis for Monte Carlo simulation studies on stochastic beams reported later in this thesis. Numerical results illustrating the formulations developed in this chapter are also presented. A satisfactory agreement of these results has been demonstrated with the corresponding results obtained from independent finite element code using normal mode expansions. The formulation of the dynamic stiffness matrix for a curved, randomly in-homogeneous, Timoshenko beam element is considered in Chapter 3. The displacement fields are discretized using the frequency dependent shape functions derived in the pre­vious chapter. These shape functions are defined with respect to a damped, uniformly parametered beam element and hence are deterministic in nature. Lagrange's equations are used to derive the 6x6 stochastic dynamic stiffness matrix of the beam element. In this formulation, the system property random fields are implicitly discretized as a set of damping and frequency dependent Weighted integrals. The results for a straight Timo- shenko beam are obtained as a special case. Numerical examples on structures made up of single curved/straight beam elements are presented. These examples also illustrate the characterization of the steady state response when excitations are modeled as stationary random processes. Issues related to ton-Gaussian features of the system in-homogeneities are also discussed. The analytical results are shown to agree satisfactorily with corresponding results from Monte Carlo simulations using 500 samples. The dynamics of structures built-up of straight and curved random Tim-oshenko beams is studied in Chapter 4. First, the global stochastic dynamic stiffness matrix is assembled. Subsequently, it is inverted for calculating the mean and variance, of the steady state stochastic response of the structure when subjected to stationary random excitations. Neumann's expansion method is adopted for the inversion of the stochastic dynamic stiffness matrix. Questions on the treatment of the beam characteris­tics as non-Gaussian random fields, are addressed. It is shown that the implementation of Neumann's expansion method and Monte-Carlo simulation method place distinc­tive demands on strategy of modeling system parameters. The Neumann's expansion method, on one hand, requires the knowledge of higher order spectra of beam properties so that the non-Gaussian features of beam parameters are reflected in the analysis. On the other hand, simulation based methods require the knowledge of the range of the stochastic variations and details of the probability density functions. The expediency of implementing Gaussian closure approximation in evaluating contributions from higher order terms in the Neumann expansion is discussed. Illustrative numerical examples comparing analytical and Monte-Carlo simulations are presented and the analytical so­lutions are found to agree favourably with the simulation results. This agreement lends credence to the various approximations involved in discretizing the random fields and inverting the global dynamic stiffness matrix. A few pointers as to how the methods developed in the thesis can be used in assessing the reliability of these structures are also given. A brief summary of contributions made in the thesis together with a few suggestions for further research are presented in Chapter 5. Appendix A describes the models of non-Gaussian random fields employed in the numerical examples considered in this thesis. Detailed expressions for the elements of the covariance matrix of the weighted integrals for the numerical example considered in Chapter 5, are presented in Appendix B; A copy of the paper, which has been ac­cepted for publication in the proceedings of IUTAM symposium on 'Nonlinearity and Stochasticity in Structural Mechanics' has been included as Appendix C.

Design based on damage tolerance concepts has become mandatory in high technology structures. These concepts are also essential for evaluating life extension of aged structures which are in service beyond originally stipulated life. Fracture analysis of such structures in the presence of single or multiple three-dimensional flaws is essential for this approach. Surface cracks are the most commonly occurring flaws and development of accurate methods of analysis for such cracks is essential for structural integrity evaluation of newly designed or aged structures. The crack fronts of these surface flaws are usually approximated mathematically to be of either part-elliptical or part-circular in geometry. In this thesis, some of the issues related to fatigue crack growth of single and multiple surface cracks are studied in detail. Here emphasis is given to the development of simple and accurate post-processing techniques to estimate stress intensity factors for surface cracks, development and/or implementation of simple numerical methods to simulate three-dimensional single and multiple cracks in fatigue and their experimental verification. Modified virtual crack closure integral (MVCCI) technique for estimation of strain energy release rates has been improved (chapter II) to deal with curved crack front and unequal elements across the crack front. The accuracy of this method is evaluated and presented in this chapter for certain benchmark surface flaw problems. The improved MVCCI is used in the investigation of interaction between multiple surface cracks in three-dimensional solids. The interaction effects are studied for both interacting and coalescing phases as observed to occur in the growth of multiple surface cracks. Extensive numerical work is performed to study the effects of various parameters such as aspect ratio, thickness ratio, interspacing on the interaction factors. These solutions are used in formulating empirical equations to estimate interaction factors. This facilitated the development of a simple semi-analytical method to study fatigue crack growth of multiple cracks. The growth of surface cracks under fatigue loading in the finite width specimens of an aero-engine superalloy has been studied experimentally (presented in chapter III). Four configurations for single semi-elliptical cracks are considered. Fatigue crack growth is simulated by two models viz. two degrees of freedom and "multi degrees of freedom with ellipse fit'. These models are sometimes referred to as semi-analytical models as the crack growth is predicted by numerical integration combining Paris equation with an empirical form of stress intensity factor solution. In order to use two degrees of freedom model for fatigue crack growth prediction of semi-elliptical cracks, empirical solution for the Ml range of geometric parameters for stress intensity factor is required for the considered configurations. The available Newman-Raju solution is useful for this purpose within a limited range of surface crack length to width (c/W) of the specimen. Based on the present finite element results, the empirical equations are developed for extended values of c/W. It is well understood that the fatigue prediction for two-dimensional crack can be improved by inclusion of crack closure effects. Usually, in semi-analytical models for growth of surface cracks under fatigue loading, the crack closure is included as a ratio of crack closure factor at surface and depth locations of semi-elliptical crack. In the present work, this ratio for the considered material of specimens is obtained by an experimental study. The difference in characteristics of preferred propagation path between semi-elliptical crack in a finite width plate and a wide plate is clearly brought out. Current crack growth predictions for most of the structures are based on the presence of only a single crack. However, in structures several cracks may initiate simultaneously within a stress critical zone and may interact depending upon their geometry, spatial location, structure geometry and mode of loading. In this work various configurations of twin semi-elliptical cracks have been studied by experiments. The beachmarks created on the specimens during experiments are used in the investigation of crack shape progression during fatigue. A three degrees of freedom crack growth model for interacting and coalescing cracks has been proposed. The experimentally determined crack shape and lives have been compared with the corresponding values from numerical simulation. The correlation of experimental results with numerical predictions was carried out through improved MVCCI for eight-noded brick elements. This has worked well in the configurations analysed. However, it is known in literature that there are benefits of using 20-noded singular elements. There could be special situations where the regular elements could fail, and singular elements could be essential. For this purpose, further development of MVCCI were carried out using 20-noded quarter-point elements (presented in chapter IV). Also a novel technique of decomposed crack closure integral (DCCI) was developed (presented in chapter V) for both regular and singular elements to represent the variation of MVCCI more accurately along the crack front. It is well known that quarter-point elements at crack front produce the required singularity at the crack tip and give accurate stress distribution with fewer degrees of freedom than conventional elements. Thus to develop more efficient post-processing tools, the MVCCI expressions are formulated for 20-noded singular quarter-point element for various assumptions regarding stress and displacement distributions in the elements across the crack front. A comprehensive study is presented (chapter IV) on MVCCI for 20-noded singular brick element including various simplified expressions for three-dimensional part-through cracks in pure and mixed-mode state of deformation of fracture. The developed MVCCI expressions are also valid for 15-noded quarter-point Penta elements. The reduction in model size can further be obtained if 12-noded three-dimensional singular element is employed at the crack front and eight-noded elements are used away from the crack front. The MVCCI expressions are also developed for 12-noded singular element and their accuracy is evaluated by numerical solutions. Presently, MVCCI, estimates the average stress intensity factor at the center of each element along the crack front. In this thesis, a Decomposed Crack Closure Integral (DCCI) is formulated to represent an assumed variation of stress intensity factor along the crack front in each element. The DCCI is formulated for 8-noded brick, 20-noded conventional brick and 20-noded singular brick elements. The numerical examples presented here deal with three-dimensional problems of patch repair technology and part-through cracks. The technique showed a major advantage for the patch repair problems where SIF variations along the crack front are of significance and large mesh sizes are computationally expensive. This along with MVCCI for 12-noded and 20-noded singular elements formed a part of the work on development of accurate and effective post-processing tools. It is expected that the present work will be helpful in damage tolerance design and assessment of aerospace structures and the experimental work performed as a part of this thesis will enhance confidence in the damage tolerance analysis. The thesis is concluded in chapter VI presenting the contributions of this thesis and projecting future lines of work possible in this area.

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