Composites are materials which cater to the present and future needs of many demanding industries, such as aerospace, as they are weight-sensitive for a given requirement of strength and stiff ness, corrosion resistant, potentially multi-functional and can be tailored according to the application. However, they are in particular difficult to join as they cannot be easily machined, without introducing damages which can eventually grow. Any structure is as strong as its weakest joint. Most of the joints belong to the category of mechanically-fastened joints and they pose enormous challenges in modeling due to contact phenomena, nonlinearity and stress concentration factors. It is therefore a necessity to construct an efficient model that would include all the relevant contact phenomena in the joints, as it has been pointed out in literature that damage typically initiates near the joint holes. The focus of this work is to describe the construction of an asymptotically-correct model using the Variational Asymptotic Method (VAM). Amongst its many potential applications, VAM is a well-established analytical tool for obtaining the stress and strain fields for beams and shells. The methodology takes advantage of the small parameter that is inherent in the problem, such as the ratio of certain characteristic dimensions of the structure. In shells and beams, VAM takes advantage of the dimension-based small parameter(s), thereby splitting the problem into 2-D + 1-D (for beams) and 1-D + 2-D (for shells), in turn offering very high computational efficiency with very little loss of accuracy compared to dimensionally unreduced 3-D models. In this work, the applicability of VAM is extended to two-dimensional (2-D) and three-dimensional (3-D) frictionless contact problems. Since a generalised VAM model for contact has not been pursued before, the `phantom0 step is adopted for both 2-D and 3-D models. The development of the present work starts with the construction of a 2-D model involving a large rectangular plate being pressed against a rigid frictionless pin. The differential equations governing the problem and the associated boundary conditions are obtained by minimizing the reduced strain energy, augmented with the appropriate gap function, by using a penalty method. The model is developed for both isotropic and orthotropic cases. The boundary value problem is solved numerically and the displacement field obtained is compared with the one obtained using commercial software (ABAQUSr) for validation at critical regions such as the contact surfaces. Banking on the validation of the 2-D model, a 3-D model with a pin and a finite annular cylinder was constructed. The strain energy for the finite cylinder was derived using geometrically exact 3-D kinematics and VAM was applied leading to the reduction in the strain energy for isotropic and orthotropic materials in rectangular and cylindrical co-ordinates. As in the 2-D case, the reduced strain energy, subject to the inequality constraint of the gap function, is minimized with respect to the displacement field and the corresponding boundary value problem is solved numerically. The displacements of the contact surface and the top surface of the annular cylinder are compared with those from ABAQUS and thus validated. The displacement fields obtained using the current 2-D and 3-D models show very good agreement with those from commercial finite element software packages. The model could be re ned further by using the gap function derived in this work and applying it to a plate model based on VAM, which could be explored in the future.

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In aerospace applications, use of laminates made of composite materials as face sheets in sandwich panels are on the rise. These composite laminates have low transverse shear and transverse normal moduli compared to the in-plane moduli. It is also seen that the corresponding transverse strength values are very low compared to the in-plane strength leading to delaminations. Further, in sandwich structures, the core is subjected to significant transverse shear stresses. Therefore the interlaminar stresses (i.e., transverse shear and normal) can govern the design of sandwich structures. As a consequence, the first step in achieving efficient designs is to develop the ability to reliably estimate interlaminar stresses. Stress analysis of the composite sandwich structures can be carried out using 3-D finite elements for each layer. Owing to the enormous computational time and resource requirements for such a model, this process of analysis is rendered inefficient. On the other hand, existing plate/shell finite elements, when appropriately chosen, can help quickly predict the 2-D displacements with reasonable accuracy. However, their ability to calculate the thickness-wise distributions of interlaminar shear and normal stresses and 3-D displacements remains as a research goal. Frequently, incremental refinements are offered over existing solutions. In this scenario, an asymptotically correct dimensional reduction from 3-D to 2-D, if possible, would serve to benchmark any ongoing research. The employment of a mathematical technique called the Variational Asymptotic Method (VAM) ensures the asymptotical correctness for this purpose. In plates and sandwich structures, it is typically possible to identify (purely from the defined material distributions and geometry) certain parameters as small compared to others. These characteristics are invoked by VAM to derive an asymptotically correct theory. Hence, the 3-D problem of plates is automatically decomposed into two separate problems (namely 1-D+2-D), which then exchange relevant information between each other in both ways. The through-the-thickness analysis of the plate, which is a 1-D analysis, provides asymptotic closed form solutions for the 2-D stiffness as well as the recovery relations (3-D warping field and displacements in terms of standard plate variables). This is followed by a 2-D plate analysis using the results of the 1-D analysis. Finally, the recovery relations regenerate all the required 3-D results. Thus, this method of developing reduced models involves neither ad hoc kinematic assumptions nor any need for shear correction factors as post-processing or curve-fitting measures. The results are most general and can be made as accurate as desired, while the procedure is computationally efficient. In the present work, an asymptotically correct plate theory is formulated for composite sandwich structures. In developing this theory, in addition to the small parameters (such as small strains, small thickness-to-wavelength ratios etc.,) pertaining to the general plate theory, additional small parameters characterizing (and specific to) sandwich structures (viz., smallness of the thickness of facial layers com-pared to that of the core and smallness of elastic material stiffness of the core in relation to that of the facesheets) are used in the formulation. The present approach also satisfies the interlaminar displacement continuity and transverse equilibrium requirements as demanded by the exact 3-D formulation. Based on the derived theory, numerical codes are developed in-house. The results are obtained for a typical sandwich panel subjected to mechanical loading. The 3-D displacements, inter-laminar normal and shear stress distributions are obtained. The results are compared with 3-D elasticity solutions as well as with the results obtained using 3-D finite elements in MSC NASTRAN®. The results show good agreement in spite of the major reduction in computational effort. The formulation is then extended for thermo-elastic deformations of a sandwich panel. This thesis is organized chronologically in terms of the objectives accomplished during the current research. The thesis is organized into six chapters. A brief organization of the thesis is presented below. Chapter-1 briefly reviews the motivation for the stress analysis of sandwich structures with composite facesheets. It provides a literature survey on the stress analysis of composite laminates and sandwich plate structures. The drawbacks of the existing anlaytical approaches as opposed to that of the VAM are brought out. Finally, it concludes by listing the main contributions of this research. Chapter-2 is dedicated to an overview of the 3-D elasticity formulation of composite sandwich structures. It starts with the 3-D description of a material point on a structural plate in the undeformed and deformed configurations. Further, the development of the associated 3-D strain field is also described. It ends with the formulation of the potential energy of the sandwich plate structure. Chapter-3 develops the asymptotically correct theory for composite sandwich plate structure. The mathematical description of VAM and the procedure involved in developing the dimensionally reduciable structural models from 3-D elasticity functional is first described. The 1-D through-the-thickness analysis procedure followed in developing the 2-D plate model of the composite sandwich structure is then presented. Finally, the recovery relations (which are one of the important results from 1-D through-the-thickness analysis) to extract 3-D responses of the structure are obtained. The developed formulation is applied to various problems listed in chapter 4. The first section of this chapter presents the validation study of the present formulation with available 3-D elasticity solutions. Here, composite sandwich plates for various length to depth ratios are correlated with available 3-D elasticity solutions as given in [23]. Lastly, the distributions of 3-D strains, stresses and displacements along the thickness for various loadings of a typical sandwich plate structure are correlated with corresponding solutions using well established 3-D finite elements of MSC NASTRAN® commerical FE software. The developed and validated formulation of composite sandwich structure for mechanical loading is extended for thermo-elastic deformations. The first sections of this chapter describes the seamless inclusion of thermo-elastic strains into the 3-D elasticity formulation. This is followed by the 1-D through-the-thickness analysis in developing the 2-D plate model. Finally, it concludes with the validation of the present formulation for a very general thermal loading (having variation in all the three co-ordinate axes) by correlating the results from the present theory with that of the corresponding solutions of 3-D finite elements of MSC NASTRAN® FE commercial software. Chapter-6 summarises the conclusions of this thesis and recommendations for future work.

Composite materials show heterogeneity at different length scales. hence concurrent multiscale analysis is the only reliable method to analyze them. But unfortunately there is no concurrent multi-scale strategy that is efficient, and accurate while addressing all kinds of problems. This lack of reliability is partly because there is no micro-mechanical model which inherently keeps all relevent global information with it. This thesis tries to fill this gap. The presented micro-mechanical model not only homogenizes the micro-structure but also keeps the global information with it. Most of the micro-mechanical models in the literature extract the Representative Volume Element (RVE) from the continuum for analysis which results in loss of information and accuracy. In the present approach also, the RVE has been extracted from the continuum but with the major difference that all the macro/meso-scopic parameters are accounted for. Five macro/meso-scopic one dimensional parameters have been defined which completely define the effect of continuum. 11 for one dimensional stretch, _1 for torsion, __ (_ = 2, 3) for bending and _33 for uniform pressurization due to the presence of the continuum. Further, the above macro/meso-scopic parameters are proven, by the asymptotic, theory to be constant at a cross section but vary, in general, over the length of the fiber. Hence, the analysis is valid for any location and is not restricted to any local domain. Three major problems have been addressed: • Homogenization and analysis of RVE without any defects • Homogenization and analysis of RVE with fiber-matrix de-bonding • Homogenization and analysis of RVE with radial matrix cracking. Variational Asymptotic Method (VAM) has been used to solve the above mentioned problems analytically. The results have been compared against standard results in the literature and against 3D FEA. At the end, results for “Radial deformation due to torsion” problem will be presented which was solved “accidentally.”

Quick and accurate predictions of equivalent properties for thin-walled composite structures are required in the preliminary design process. Existing literature provides analytical solutions to some structures but is limited to particular cases. No unified approach exists to tackle homogenization of thin-walled structures such as beams, plates, or three-dimensional structures using the thin-walled approximation. In this work, a unified approach is proposed to obtain equivalent properties for beams, plates, and three-dimensional structures for thin-walled composite structures using mechanics of structure genome. The adopted homogenization technique interprets the unit cell associated with the composite structures as an assembly of plates, and the overall strain energy density of the unit cell as a summation of the plate strain energies of these individual plates. The variational asymptotic method is then applied to drop all higher-order terms and the remaining energy is minimized with respect to the unknown fluctuating functions. This has been done by discretizing the two-dimensional unit cell into one-dimensional frame elements in a finite element description. This allows the handling of structures with different levels of complexities and internal geometry within a general framework. Comparisons have been made with other works to show the advantages which the proposed model offers over other methods.

In this thesis, we develop some inferential results for one-shot device testing data analysis. These extend and generalize existing methods in the literature. First, a competing-risk model is introduced for one-shot testing data under accelerated life-tests. One-shot devices are products which will be destroyed immediately after use. Therefore, we can observe only a binary status as data, success or failure, of such products instead of its lifetime. Many one-shot devices contain multiple components and failure of any one of them will lead to the failure of the device. Failed devices are inspected to identify the specific cause of failure. Since the exact lifetime is not observed, EM algorithm becomes a natural tool to obtain the maximum likelihood estimates of the model parameters. Here, we develop the EM algorithm for competing exponential and Weibull cases. Second, a semi-parametric approach is developed for simple one-shot device testing data. Semi-parametric estimation is a model that consists of parametric and non-parametric components. For this purpose, we only assume the hazards at different stress levels are proportional to each other, but no distributional assumption is made on the lifetimes. This provides a greater flexibility in model fitting and enables us to examine the relationship between the reliability of devices and the stress factors. Third, Bayesian inference is developed for one-shot device testing data under exponential distribution and Weibull distribution with non-constant shape parameters for competing risks. Bayesian framework provides statistical inference from another perspective. It assumes the model parameters to be random and then improves the inference by incorporating expert's experience as prior information. This method is shown to be very useful if we have limited failure observation wherein the maximum likelihood estimator may not exist. The thesis proceeds as follows. In Chapter 2, we assume the one-shot devices to have two components with lifetimes having exponential distributions with multiple stress factors. We then develop an EM algorithm for developing likelihood inference for the model parameters as well as some useful reliability characteristics. In Chapter 3, we generalize to the situation when lifetimes follow a Weibull distribution with non-constant shape parameters. In Chapter 4, we propose a semi-parametric model for simple one-shot device test data based on proportional hazards model and develop associated inferential results. In Chapter 5, we consider the competing risk model with exponential lifetimes and develop inference by adopting the Bayesian approach. In Chapter 6, we generalize these results on Bayesian inference to the situation when the lifetimes have a Weibull distribution. Finally, we provide some concluding remarks and indicate some future research directions in Chapter 7. / Thesis / Doctor of Philosophy (PhD)

This work aims at dimensional reduction of nonlinear material models in an asymptotically accurate manner. The three-dimensional(3-D) nonlinear material models considered include isotropic, orthotropic and dielectric compressible hyperelastic material models. Hyperelastic materials have potential applications in space-based inflatable structures, pneumatic membranes, replacements for soft biological tissues, prosthetic devices, compliant robots, high-altitude airships and artificial blood pumps, to name a few. Such structures have special engineering properties like high strength-to-mass ratio, low deflated volume and low inflated density. The majority of these applications imply a thin shell form-factor, rendering the problem geometrically nonlinear as well. Despite their superior engineering properties and potential uses, there are no proper analysis tools available to analyze these structures accurately yet efficiently. The development of a unified analytical model for both material and geometric nonlinearities encounters mathematical difficulties in the theory but its results have considerable scope. Therefore, a novel tool is needed to dimensionally reduce these nonlinear material models. In this thesis, Prof. Berdichevsky’s Variational Asymptotic Method(VAM) has been applied rigorously to alleviate the difficulties faced in modeling thin shell structures(made of such nonlinear materials for the first time in the history of VAM) which inherently exhibit geometric small parameters(such as the ratio of thickness to shortest wavelength of the deformation along the shell reference surface) and physical small parameters(such as moderate strains in certain applications). Saint Venant-Kirchhoff and neo-Hookean 3-D strain energy functions are considered for isotropic hyperelastic material modeling. Further, these two material models are augmented with electromechanical coupling term through Maxwell stress tensor for dielectric hyperelastic material modeling. A polyconvex 3-D strain energy function is used for the orthotropic hyperelastic model. Upon the application of VAM, in each of the above cases, the original 3-D nonlinear electroelastic problem splits into a nonlinear one-dimensional (1-D) through-the-thickness analysis and a nonlinear two-dimensional(2-D) shell analysis. This greatly reduces the computational cost compared to a full 3-D analysis. Through-the-thickness analysis provides a 2-D nonlinear constitutive law for the shell equations and a set of recovery relations that expresses the 3-D field variables (displacements, strains and stresses) through thethicknessintermsof2-D shell variables calculated in the shell analysis (2-D). Analytical expressions (asymptotically accurate) are derived for stiffness, strains, stresses and 3-D warping field for all three material types. Consistent with the three types of 2-D nonlinear constitutive laws,2-D shell theories and corresponding finite element programs have been developed. Validation of present theory is carried out with a few standard test cases for isotropic hyperelastic material model. For two additional test cases, 3-Dfinite element analysis results for isotropic hyperelastic material model are provided as further proofs of the simultaneous accuracy and computational efficiency of the current asymptotically-correct dimensionally-reduced approach. Application of the dimensionally-reduced dielectric hyperelastic material model is demonstrated through the actuation of a clamped membrane subjected to an electric field. Finally, the through-the-thickness and shell analysis procedures are outlined for the orthotropic nonlinear material model.

A device that is accompanied by an irreversible chemical reaction or physical destruction and could no longer function properly after performing its intended function is referred to as a one-shot device. One-shot device test data differ from typical data obtained by measuring lifetimes in standard life-tests. Due to the very nature of one-shot devices, actual lifetimes of one-shot devices under test cannot be observed, and they are either left- or right-censored. In addition, a one-shot device often has multiple components that could cause the failure of the device. The components are coupled together in the manufacturing process or assembly, resulting in the failure modes possessing latent heterogeneity and dependence. Frailty models enable us to describe the influence of common, but unobservable covariates, on the hazard function as a random effect in a model and also provide an easily understandable interpretation. In this thesis, we develop some inferential results for one-shot device testing data with gamma frailty model. We first develop an efficient expectation-maximization (EM) algorithm for determining the maximum likelihood estimates of model parameters of a gamma frailty model with exponential lifetime distributions for components based on one-shot device test data with multiple failure modes, wherein the data are obtained from a constant-stress accelerated life-test. The maximum likelihood estimate of the mean lifetime of $k$-out-of-$M$ structured one-shot devices under normal operating conditions is also presented. In addition, the asymptotic variance–covariance matrix of the maximum likelihood estimates is derived, which is then used to construct asymptotic confidence intervals for the model parameters. The performance of the proposed inferential methods is finally evaluated through Monte Carlo simulations and then illustrated with a numerical example. A gamma frailty model with Weibull baseline hazards is considered next for fitting one-shot device testing data. The Weibull baseline hazards enable us to analyze time-varying failure rates more accurately, allowing for a deeper understanding of the dynamic nature of system's reliability. We develop an EM algorithm for estimating the model parameters utilizing the complete likelihood function. A detailed simulation study evaluates the performance of the Weibull baseline hazard model with that of the exponential baseline hazard model. The introduction of shape parameters in the component's lifetime distribution within the Weibull baseline hazard model offers enhanced flexibility in model fitting. Finally, Bayesian inference is then developed for the gamma frailty model with exponential baseline hazard for one-shot device testing data. We introduce the Bayesian estimation procedure using Markov chain Monte Carlo (MCMC) technique for estimating the model parameters as well as for developing credible intervals for those parameters. The performance of the proposed method is evaluated in a simulation study. Model comparison between independence model and the frailty model is made using Bayesian model selection criterion. / Thesis / Candidate in Philosophy

This thesis considers new methods that exploit recent developments in computer technology to address three extant problems in the area of Finance and Econometrics. The problem of Asian option pricing has endured for the last two decades in spite of many attempts to find a robust solution across all parameter values. All recently proposed methods are shown to fail when computations are conducted using standard machine precision because as more and more accuracy is forced upon the problem, round-off error begins to propagate. Using recent methods from numerical analysis based on multi-precision arithmetic, we show using the Mathematica platform that all extant methods have efficacy when computations use sufficient arithmetic precision. This creates the proper framework to compare and contrast the methods based on criteria such as computational speed for a given accuracy. Numerical methods based on a deformation of the Bromwich contour in the Geman-Yor Laplace transform are found to perform best provided the normalized strike price is above a given threshold; otherwise methods based on Euler approximation are preferred. The same methods are applied in two other contexts: the simulation of stable distributions and the computation of unit root densities in Econometrics. The stable densities are all nested in a general function called a Fox H function. The same computational difficulties as above apply when using only double-precision arithmetic but are again solved using higher arithmetic precision. We also consider simulating the densities of infinitely divisible distributions associated with hyperbolic functions. Finally, our methods are applied to unit root densities. Focusing on the two fundamental densities, we show our methods perform favorably against the extant methods of Monte Carlo simulation, the Imhof algorithm and some analytical expressions derived principally by Abadir. Using Mathematica, the main two-dimensional Laplace transform in this context is reduced to a one-dimensional problem.