The mixing of compounds in a highly viscous medium is important in many industrial settings; from food processing to the manufacturing of rocket fuel and drugs. Experts in mixing have long been aware of how things become mixed in a nonturbulent flow, but there has been little quantitative analysis of such mixing processes. As recent developments in chaos theory have found their way into the engineering literature, there have been some attempts to apply these ideas toward numerically quantifying nonturbulent mixing processes. Chaos theory is a new name for an old subject in mathematics, dynamical systems theory which includes ergodic theory. By examining the older literature of ergodic theory, one can determine what is necessary to quantify nonturbulent mixing processes. This has led to the methods which are suggested in this dissertation. After discussing some principles of ergodic theory, the design of a bladeless mixer is presented. The philosophy of this design is to adopt an abstract mathematically mixing system around which to design and build an actual machine. Ergodic theory is then used to develop methods for quantifying nonturbulent mixing processes by both experimental and numerical means. These methods are then applied to the bladeless mixer.

<p>Continuous-fiber ceramic matrix composites (CMCs) are of interest as hot-section components in gas turbine engines due to their refractoriness and low density relative to metallic alloys. In service, CMCs will be subjected to spatially inhomogeneous temperature and stress fields. Robust tools that enable prediction of deformation and fracture under these conditions are therefore required for component design and analysis. Such tools are presently lacking. The present work helps to address this deficiency by developing models for CMC mechanical behavior at two length scales: that of the constituents and that of the components. Problems of interest are further divided into two categories: &lsquo;1-D loadings,&rsquo; in which the stresses are aligned with the fiber axes, and &lsquo;2-D loadings,&rsquo; in which the stress state is more general. </p><p> For the former class of problems, the major outstanding issue is material fracture, not deformation. A fracture criterion based on the attainment of a global load maximum is developed, which yields results for pure bending of CMCs in reasonable agreement with available experimental data. For the latter class of problems, the understanding of both the micro-scale and macro-scale behavior is relatively immature. An approach based upon analysis of a unit cell (a single fiber surrounded by a matrix jacket) is pursued. Stress fields in the constituents of the composite are estimated using analytical models, the accuracy of which is confirmed using finite element analysis. As part of a fracture mechanics analysis, these fields enable estimation of the steady-state matrix cracking stress for arbitrary in-plane loading of a unidirectional ply. While insightful at the micro-scale, unit cell models are difficult to extend to coarser scales. Instead, material deformation is typically predicted using phenomenological constitutive models. One such model for CMC laminates is investigated and found to predict material instability where none should exist. Remedies to the model to correct this deficiency are proposed; the remediated model is subsequently utilized in conjunction with an analytical model to probe stress fields adjacent to holes and notches in CMC panels. However, even the revised model is incapable of capturing the range of experimental behavior reported for CMCs with both stiff and compliant matrices. To ameliorate this deficiency, a new elastic-plastic constitutive model is developed. It extends the deformation theory of plasticity from metals to CMCs, and its predictions of near-notch strain fields in an open-hole tension test compare favorably to strains measured using digital image correlation. Based on these developments, future experimental and modeling work is proposed. With respect to the latter, cohesive interface simulations seem particularly suited for capturing multiple interacting damage mechanisms at multiple length scales in a physically sensible manner. In principle, they can function as virtual tests, guiding both engineering design and materials development. </p>