Browsing by Subject "microstructure"
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Item Analysis of wave motion in 1D waveguides with microstructure(2020-12) Panahandeh-Shahraki, DanialThe main objectives of this dissertation include: (I) low-frequency homogenization of the wave equation in Rd with non-uniformly oscillating coefficients, (II) low-frequency solution of boundary value problem for 1D quasi-periodic waveguides, (III) solution of boundary value problem for 1D purely periodic waveguides at arbitrary frequency, as described below. Effective field equation. The focus of this work is dispersive, second-order effective model describing the low-frequency wave motion in heterogeneous (e.g. functionally- graded) media endowed with periodic microstructure. For this class of quasi-periodic medium variations, homogenization of the scalar wave equation in R^d, d >= 1 is pursued within the framework of multiple scales expansion. When either d = 1 or d = 2, this model problem bears direct relevance to the description of (anti-plane) shear waves in elastic solids. By adopting the lengthscale of microscopic medium fluctuations as the perturbation parameter, the germane low-frequency behavior is synthesized via a fourth- order differential equation (with smoothly varying coefficients) governing the mean wave motion in the medium, where the effect of microscopic heterogeneities is upscaled by way of the so-called cell functions. Effective boundary value problem. In an effort to demonstrate the relevance of the proposed analysis, presented in objective (I), toward solving boundary value problems (deemed to be the ultimate goal of most homogenization studies), effective boundary conditions are also developed, up to the second order of asymptotic approximation, applicable to one-dimensional (1D) shear wave motion in a macroscopically heterogeneous solid with periodic microstructure. The analysis is illustrated numerically in 1D by considering (i) low-frequency wave dispersion, (ii) mean-field homogenized description of the shear waves propagating in a finite domain, and (iii) full-field homogenized description thereof. In contrast to (i) where the overall wave dispersion appears to be fairly well described by the leading-order model, the results in (ii) and (iii) demonstrate the critical role that higher-order corrections may have in approximating the actual waveforms in quasi-periodic media. From d’Alembert to Bloch and back. Regarding the third objective, a simple semi-analytical model is developed for solving boundary value problems in one- dimensional (1D) periodic waveguides at arbitrary frequency. When the boundary value problem is well-posed in that the excitation frequency does not match resonant frequency of the waveguide, any solution that satisfies the field equation and boundary conditions is the (unique) solution of the problem. This motivates this study to seek the solution in the form of Bloch waves that by design solve the wave equation with periodic coefficients. For a given excitation frequency this assumption leads to a quadratic eigenvalue problem in terms of the wavenumber k which, in turn, produces a d’Alembert - type solution in terms of the left- and right- “propagating” waves. In this vein, the solution of a boundary value problem is obtained by computing the “left” and “right” amplitudes of the Bloch waves from the prescribed boundary conditions. For completeness, the situations are considered when the driving frequency is (i) inside a passband, (ii) inside a band gap, (iii) at the edge of a band gap (the so-called exceptional points), and (iv) at a crossing point between dispersion branches (repeated eigenfrequencies). From simulations, it is found that the semi-analytical solution (expressed in terms of either propagating or evanescent waves) is computationally very effective and reproduces the numerical results with high fidelity. Consistent with related studies, it is reported that the solution of the boundary value problem undergoes sharp transition in neighborhood of exceptional points. Perhaps the most remarkable result of this study is the find- ing that the resonance of a waveguide can be induced by adjusting the “microscopic” location of the support points so they coincide with the nodes of the Bloch eigenfunctions (if any) at given frequency. Vice versa, one can minimize the vibration amplitude (for given boundary condition) by selecting the micro-location of the support points near peaks of the germane Bloch eigenfunctions. This opens the door toward design of adaptive periodic systems whose performance can be tuned by adjusting the boundary conditions as opposed to altering the properties of the periodic medium. Applications of our initial study may include adaptive energy harvesting for sensing purposes and vibration isolation. The extension of this work to nonlinear periodic systems, those with internal resonators (metamaterials), and more complex physical systems such as beams is envisioned.Item Optical micrographs of olivine + melt samples deformed in torsion(2018-02-09) Qi, Chao; Kohlstedt, David L; qixxx063@umn.edu; Qi, ChaoThis data set contains high-resolution micrographs of the transverse sections of partially molten samples deformed in torsion. These micrographs present the distribution of melt induced by deformation, which is a test of the melt segregation processes predicted by the two-phase flow theory incorporating viscous anisotropy.Item A study of microfabric and structures in glacial ice: insights from Storglaciären, northern Sweden(2021-06) Monz, MorganBetter understanding of the flow of ice is essential for helping to predict and adapt to the global mean sea level rise in the coming decades and centuries. Flow produces internal structures and fabrics in glaciers that in turn influence the flow. Documenting and understanding the internal structure of glaciers is thus important for understanding how glaciers behave and respond to changing conditions. This thesis focuses on establishing the significance of commonly described but enigmatic crystallographic ice fabrics and associated structures, with specific examples taken from Storglaciären, a small, polythermal valley glacier in northern Sweden. Characteristic microstructures in deformed ice in valley glaciers and at the base of ice sheets include large, irregularly shaped interlocking grains with no clear shape-preferred orientation, little internal distortion, and c-axis crystallographic preferred orientations (CPOs) defined by several maxima. Such fabrics develop in warm ice (T> -10 o C). They are difficult to quantify and interpret in part because the large grain size and interlocking texture creates issues with quantifying the true size and shape of individual crystals, and thus with determining a representative CPO. All but one previous study have analyzed these fabrics using techniques that can only measure c-axis orientations and thus partial CPOs. A newly developed sample preparation method allows for the first use of electron backscatter diffraction (EBSD) on such coarse-grained ice, which then provides high accuracy measurements of full crystallographic orientations (c- and a-axis measurements). Both EBSD and the older Rigsby stage optical technique (c-axis only) are used in this study and this allows for comparison of the two methods and the results of this study with the results of previous work. In all samples analyzed from Storglaciären, the ice has been entirely recrystallized by dynamic recrystallization accommodated by grain boundary migration. The degree to which a grain is irregularly shaped is related to the density of bubbles, which are a secondary phase and which act to slow or stop grain boundary migration. There is no consistent angular relationship between the orientations of the a-axes and c-axes of two adjacent grains that might indicate a twin relationship and thus provide an explanation for the multimaxima c-axis fabric, as had been previously suggested. In nearly all cases c-axis maxima themselves lie in a cluster at a high angle to a plane of high shear stress or shear strain and to the foliation plane. As the number of grains included in a fabric is increased by combining data from several samples, thus creating a more representative sample, the multimaxima nature of the patterns diminishes, and in some cases the fabrics can be understood as being consistent with a simpler interpretation than previously proposed, specifically with fabrics developed under simple shear. Additionally, small bands of ice that are entirely pinned by bubbles and characterized by finer grains display a more clearly defined CPO reflecting simple shear. These may represent zones of localized high strain. On the microscale as on the macroscale, many structures in ice are comparable to those developed in rocks undergoing high temperature deformation. These include fractures, foliation, and folds. In many glaciers, including Storglaciären, ridges of basally-derived debris present on the surface in the ablation zone are interpreted as having being emplaced along thrust faults, even though thrust faulting has not been demonstrated (except in one surging glacier) and should be mechanically inhibited in ice at natural strain rates. An alternative hypothesis is presented that calls on a combination of high and variable subglacial water pressure and tensile fracturing, based on field observations, CPO, sediment and isotopic analyses, supported by simple modeling. The debris ridges on the surface were likely incorporated into the ice by refreezing processes or injection into fractures opened during times of high water pressure near the slip/no slip (frozen near the toe) transition. Once the debris is incorporated into the glacier, the bands of debris features are transported forwards and upwards in the ice due to basal shear and longitudinal compression and vertical extension in the ablation zone.