Reinforced cylindrical shells are used in numerous industries; common examples include undersea vehicles and industrial piping. Current models typically incorporate approximate theories to determine shell behavior, which have limitations in terms of both thickness and frequency. In addition, many applications feature coatings on the shell surface that normally have thicknesses which must also be considered. To increase the fidelity of such systems, this work develops an analytical model of an elastic cylindrical shell featuring periodically spaced ring stiffeners with an acoustic coating applied to the outer surface. There is an external fluid environment. Beginning with the equations of elasticity for a solid, spatial-domain displacement field solutions are produced incorporating unknown wave propagation coefficients. These fields are used to determine stresses at the boundaries of the shell and coating, which are then coupled with stresses from the stiffeners and fluid. The stress boundary conditions contain double-index infinite summations, which are decoupled, truncated, and recombined into a global matrix equation. The solution to this global equation results in the displacement responses of the system as well as the scattered pressure field. Two distinct loadings are considered: a ring loading and an incident acoustic wave. Thin-shell reference models are used for validation, and the acoustic response of the system is examined. It is shown that the reinforcing ribs and acoustic coating have a considerable effect on system behavior. / Master of Science / Reinforced cylindrical shells are used in numerous industries; common examples include undersea vehicles and industrial piping. Current models typically incorporate approximate theories to determine shell behavior, which have limitations in terms of both thickness and frequency. In addition, many applications feature coatings on the shell surface that normally have thicknesses which must also be considered. To increase the fidelity of such systems, this work develops an analytical model of an elastic cylindrical shell featuring periodically spaced ring stiffeners with an acoustic coating applied to the outer surface. There is an external fluid environment. Beginning with elastic equations of motion for a solid, the displacements of the system can be found. These displacements are used to determine stresses at the boundaries of the shell and coating, which are then coupled with stresses from the stiffeners and fluid. Techniques are used to transform the stress boundary conditions into a large matrix equation, and the solution to this global equation results in the displacement responses of the system as well as the scattered pressure field. Two distinct loadings are considered: a ring loading and an incident acoustic wave. Thin-shell reference models are used for validation, and the acoustic response of the system is examined. It is shown that the reinforcing ribs and acoustic coating have a considerable effect on system behavior.

This thesis focuses on the effects of slip-slip, slip-twin, and slip-precipitates interactions on strain hardening, with the intent of developing comprehensive modelling capabilities enabling to investigate unit processes and their collective effects up to the macroscopic response. To this end, the modelling strategy adopted in this work relies on a two-way exchange of information between predictions obtained by discrete dislocation dynamics (DDD) simulations and crystal plasticity laws informed by DDD. At the scale of lattice defects, a DDD tool enabling simulations on any crystalline structure is developed to model dislocation-dislocation, dislocation-twin and dislocation-particles interactions. The tool is first used to quantify the collective effect and strength of dislocation-dislocation interactions on latent-hardening, especially in the case of pure Mg. With regards to slip-twin interactions, a transmission mechanism is implemented in the DDD framework so as to investigate the collective effects of dislocation transmission across a twin-boundary. With respect to slip-particles interactions, an efficient novel DDD approach based on a Fast Fourier Transform (FFT) technique is developed to include the field fluctuations related to elastic heterogeneities giving rise to image forces on dislocation lines. In addition, the DDD-FFT approach allows for the efficient treatment of anisotropic elasticity, thereby paving the way towards performing DDD simulations in low-symmetry polycrystals. The information extracted from the collective dislocation interactions are then passed to a series of constitutive models, and later used to quantify their effects at the scale of the polycrystal. For such purpose, a constitutive framework capable of receiving information from lower scales and establishing a direct connection with DDD simulations is notably developed.

In this dissertation we investigate the behavior of radially symmetric non-Euclidean plates of thickness t with constant negative Gaussian curvature. We present a complete study of these plates using the Föppl-von Kármán and Kirchhoff reduced theories of elasticity. Motivated by experimental results, we focus on deformations with a periodic profile. For the Föppl-von Kármán model, we prove rigorously that minimizers of the elastic energy converge to saddle shaped isometric immersions. In studying this convergence, we prove rigorous upper and lower bounds for the energy that scale like the thickness t squared. Furthermore, for deformation with n-waves we prove that the lower bound scales like nt² while the upper bound scales like n²t². We also investigate the scaling with thickness of boundary layers where the stretching energy is concentrated with decreasing thickness. For the Kichhoff model, we investigate isometric immersions of disks with constant negative curvature into R³, and the minimizers for the bending energy, i.e. the L² norm of the principal curvatures over the class of W^2,2 isometric immersions. We show the existence of smooth immersions of arbitrarily large geodesic balls in H² into R³. In elucidating the connection between these immersions and the nonexistence/ singularity results of Hilbert and Amsler, we obtain a lower bound for the L^∞ norm of the principal curvatures for such smooth isometric immersions. We also construct piecewise smooth isometric immersions that have a periodic profile, are globally W^2,2, and numerically have lower bending energy than their smooth counterparts. The number of periods in these configurations is set by the condition that the principal curvatures of the surface remain finite and grow approximately exponentially with the radius of the disc.

<p>Implemented on an intracardiac echocardiography transducer, acoustic radiation force methods may provide a useful means of characterizing the heart's elastic properties. Elasticity imaging may be of benefit for diagnosis and characterization of infarction and heart failure, as well as for guidance of ablation therapy for the treatment of arrhythmias. This thesis tests the hypothesis that with appropriately designed imaging sequences, intracardiac acoustic radiation force impulse (ARFI) imaging and shear wave elasticity imaging (SWEI) are viable tools for quantification of myocardial elasticity, both temporally and spatially. Multiple track location SWEI (MTL-SWEI) is used to show that, in healthy in vivo porcine ventricles, shear wave speeds follow the elasticity changes with contraction and relaxation of the myocardium, varying between 0.9 and 2.2 m/s in diastole and 2.6 and 5.1 m/s in systole. Infarcted tissue is less contractile following infarction, though not unilaterally stiffer. Single-track-location SWEI (STL-SWEI) is proven to provide suppression of speckle noise and enable improved resolution of structures smaller than 2 mm in diameter compared to ARFI and MTL-SWEI. Contrast to noise ratio and lateral edge resolution are shown to vary with selection of time step for ARFI and arrival time regression filter size for STL-SWEI and MTL-SWEI. </p><p>In 1.5 mm targets, STL-SWEI achieves alternately the tightest resolution (0.3 mm at CNR = 3.5 for a 0.17 mm filter) and highest CNR (8.5 with edge width = 0.7 mm for a 0.66 mm filter) of the modalities, followed by ARFI and then MTL-SWEI.</p><p>In larger, 6 mm targets, the CNR-resolution tradeoff curves for ARFI and STL-SWEI overlap for ARFI time steps up to 0.5 ms and kernels $\leq$1 mm for STL-SWEI. STL-SWEI can operate either with a 25 dB improvement over MTL-SWEI in CNR at the same resolution, or with edge widths 5$\times$ as narrow at equivalent CNR values, depending on the selection of regression filter size. Ex vivo ablations are used to demonstrate that ARFI, STL-SWEI and MTL-SWEI each resolve ablation lesions between 0.5 and 1 cm in diameter and gaps between lesions smaller than 5 mm in 3-D scans. Differences in contrast, noise, and resolution between the modalities are discussed. All three modalities are also shown to resolve ``x''-shaped ablations up to 22 mm in depth with good visual fidelity and correspondence to surface photographs, with STL-SWEI providing the highest quality images. Series of each type of image, registered using 3-D data from an electroanatomical mapping system, are used to build volumes that show ablations in in vivo canine atria. In vivo images are shown to be subject to increased noise due to tissue and transducer motion, and the challenges facing the proposed system are discussed. Ultimately, intracardiac acoustic radiation force methods are demonstrated to be promising tools for characterizing dynamic myocardial elasticity and imaging radiofrequency ablation lesions.</p> / Dissertation

The method of steepest descent is used to minimize typical functionals from elasticity.

In this thesis, we applied local analysis tools (eigenvalue and eigenvalue elasticity analysis, global function elasticity/sensitivity analysis), and global analysis tools (deriving the location and stability of fixed points) to both aggregate and individual-level dynamic models of infectious diseases. We sought to use these methods to gain insight into the models and to evaluate the use of these methods to study their short-term and long-term dynamics and the influences of arameters on the models.<p> We found that eigenvalues are effective for understanding short-term behaviours of a nonlinear system, but less effective in providing insights of the long-term impacts of a parameter change on its behaviours. In term of disease control, local changes of behaviours, yielded from the changes of parameters based on eigenvalue elasticity, are able to alter behaviours in a short-term, especially in the period of a disease outbreak. While eigenvalue elasticity analysis can be helpful for understanding the impact of parameter changes for simple aggregate models, such analyses prove unwieldy and complicated, particularly for models with large number of state variables; and easily fall prey to eigenvalue multiplicity problems for large individual-based models, and istracting artifacts associated with small denominators. In response to these concerns, we introduced other local methods (global function elasticity/sensitivity analyses) that capture many of the advantages of eigenvalue elasticity methods with much greater simplicity. Unfortunately, parameter changes guided by these local analysis techniques are often insufficient to alter behaviours in the longer-term, such as when system behaviours approach stable endemic equilibria. By contrast, the global analytic tools, such as fixed point location and stability analysis, are effective for providing insights into the global behaviours of disease spread in the long-term, as well as their dependence on parameters. Using all of the above analysis as a toolset, we gained some possible insights into combination of local and global approaches. Choice of applying local or global analysis tools to infectious disease models is dependent on the specific target of policy makers as well as model type.

In this thesis, we applied local analysis tools (eigenvalue and eigenvalue elasticity analysis, global function elasticity/sensitivity analysis), and global analysis tools (deriving the location and stability of fixed points) to both aggregate and individual-level dynamic models of infectious diseases. We sought to use these methods to gain insight into the models and to evaluate the use of these methods to study their short-term and long-term dynamics and the influences of arameters on the models.<p> We found that eigenvalues are effective for understanding short-term behaviours of a nonlinear system, but less effective in providing insights of the long-term impacts of a parameter change on its behaviours. In term of disease control, local changes of behaviours, yielded from the changes of parameters based on eigenvalue elasticity, are able to alter behaviours in a short-term, especially in the period of a disease outbreak. While eigenvalue elasticity analysis can be helpful for understanding the impact of parameter changes for simple aggregate models, such analyses prove unwieldy and complicated, particularly for models with large number of state variables; and easily fall prey to eigenvalue multiplicity problems for large individual-based models, and istracting artifacts associated with small denominators. In response to these concerns, we introduced other local methods (global function elasticity/sensitivity analyses) that capture many of the advantages of eigenvalue elasticity methods with much greater simplicity. Unfortunately, parameter changes guided by these local analysis techniques are often insufficient to alter behaviours in the longer-term, such as when system behaviours approach stable endemic equilibria. By contrast, the global analytic tools, such as fixed point location and stability analysis, are effective for providing insights into the global behaviours of disease spread in the long-term, as well as their dependence on parameters. Using all of the above analysis as a toolset, we gained some possible insights into combination of local and global approaches. Choice of applying local or global analysis tools to infectious disease models is dependent on the specific target of policy makers as well as model type.

In this study, we apply "the equilibrium price" to investigate the efficiency and the elasticity of Taiwan securities trading market. The "the equilibrium price" of each transaction are used to represent the true price of the security. The intra-daily tick-by-tick data of the Taiwan security market is used to obtain the equilibrium prices. Empirical transaction of the two companies Uni-President Enterprises Corporation and Formosa Plastics Corporation are studied. Time-series models of the equilibrium price and the transaction price are established. The time lengths returning to the equilibrium status are also studied, called the efficiency time. Based on the results, we discuss the efficiency of the two stocks. In order to understand the impact of the efficiency time, linear regression models of the efficiency time are built. Furthermore, the variance ratios of the two stocks are also investigated to study their market efficiency. Finally, the elasticity of demand and the elasticity of supply are studied and their Markov chain models are established. The results show that the two companies stay more time in the inelastic states.

In this research, we study two different geometric approaches, namely, the discrete exterior calculus and differential complexes, for developing numerical schemes for linear and nonlinear elasticity. Using some ideas from discrete exterior calculus (DEC), we present a geometric discretization scheme for incompressible linearized elasticity. After characterizing the configuration manifold of volume- preserving discrete deformations, we use Hamilton’s principle on this configuration manifold. The discrete Euler-Lagrange equations are obtained without using Lagrange multipliers. The main difference between our approach and the mixed finite element formulations is that we simultaneously use three different discrete spaces for the displacement field. We test the efficiency and robustness of this geometric scheme using some numerical examples. In particular, we do not see any volume locking and/or checkerboarding of pressure in our numerical examples. This suggests that our choice of discrete solution spaces is compatible. On the other hand, it has been observed that the linear elastostatics complex can be used to find very efficient numerical schemes. We use some geometric techniques to obtain differential complexes for nonlinear elastostatics. In particular, by introducing stress functions for the Cauchy and the second Piola-Kirchhoff stress tensors, we show that 2D and 3D nonlinear elastostatics admit separate kinematic and kinetic complexes. We show that stress functions corresponding to the first Piola-Kirchhoff stress tensor allow us to write a complex for 3D nonlinear elastostatics that similar to the complex of 3D linear elastostatics contains both the kinematics an kinetics of motion. We study linear and nonlinear compatibility equations for curved ambient spaces and motions of surfaces in R3. We also study the relationship between the linear elastostatics complex and the de Rham complex. The geometric approach presented in this research is crucial for understanding connections between linear and nonlinear elastostatics and the Hodge Laplacian, which can enable one to convert numerical schemes of the Hodge Laplacian to those for linear and possibly nonlinear elastostatics.

The primary objective of this research program was to investigate the effective modulus of elasticity and shear modulus of brick masonry walls under lateral load, and to to justify using the Jaeger and Mufti method to calculate the effective modulus of elasticity and shear modulus of brick masonry walls. The experimental program involved the testing of three unreinforced brick masonry walls under in-plane and vertical loads. Linear Variable Differential Transducers were used to record the horizontal and vertical displacements of the walls. The experimental results were used to evaluate the modulus of elasticity and the shear modulus of walls under flexure. The experimental results were compared to the finite element analysis results. It was found that the finite element analysis yields similar results to the experimental results. It was also found that the Jaeger and Mufti method to calculate effective modulus of elasticity and shear modulus of brick masonry walls is effective for design purposes.