The theory and computation of line defects are discussed in the context of both solids and liquid crystals. This dissertation includes four parts. The Generalized Disclination theory is discussed and applied to numerous interfacial and bulk line defect problems. An augmented Oseen-Frank energy as well as a novel 2D-model is proposed and demonstrated for disclination dynamics in liquid crystal. A model based on kinematics and thermodynamics is devised to predict tactoid dynamics during the process of the isotropic-nematic phase transition in LCLC. In the first part of the thesis, the utility of the notion of generalized disclinations in materials science is discussed within the physical context of modeling interfacial and bulk line defects. The Burgers vector of a disclination dipole in linear elasticity is derived, clearly demonstrating the equivalence of its stress field to that of an edge dislocation. An explicit formula for the displacement jump of a single localized composite defect line in terms of given g.disclination and dislocation strengths is deduced based on the Weingarten theorem for g.disclination theory at finite deformation. The Burgers vector of a g.disclination dipole at finite deformation is also derived. In the second part, a numerical method is developed to solve for the stress and distortion fields of g.disclination systems. Problems of small and finite deformation theory are considered. The fields of various line defects and grain/phase boundary problems are approximated. It is demonstrated that while the far-field topological identity of a dislocation of appropriate strength and a disclinationdipole plus a slip dislocation comprising a disconnection are the same, the latter microstructure is energetically favorable. This underscores the complementary importance of all of topology, geometry, and energetics (plus kinetics) in understanding defect mechanics. It is established that finite element approximations of fields of interfacial and bulk line defects can be achieved in a systematic and routine manner, thus contributing to the study of intricate defect microstructures in the scientific understanding and predictive design of materials. In the third part, nonsingular disclination dynamics in a uniaxial nematic liquid crystal is modeled within a mathematical framework where the kinematics is a direct extension of the classical way of identifying these line defects with singularities of a unit vector field representing the nematic director. We devise a natural augmentation of the Oseen-Frank energy to account for physical situations where infinite director gradients have zero associated energy cost, as would be necessary for modeling half-integer strength disclinations within the framework of the director theory. A novel 2D-model of disclination dynamics in nematics is proposed, which is based on the extended Oseen-Frank energy and takes into account thermodynamics and the kinematics of conservation of defect topological charge. We validate this model through computations of disclination equilibria, annihilation, repulsion, and splitting. In the fourth part, the isotropic-nematic phase transition in chromonic liquid crystals is studied. We simulate such tactoid equilibria and dynamics with a model using degree of order, a variable length director as state descriptors, and an interfacial descriptor. We introduce an augmented Oseen-Frank energy, with non-convexity in both interfacial energy and the dependence of the energy on the degree of order. A strategy is devised based on continuum kinematics and thermodynamics. The model is used to predict tactoid dynamics during the process of phase transition. We reproduce observed behaviors in experiments and perform an experimentally testable parametric study of the effect of bulk elastic and tactoid interfacial energy constants on the interaction of interfacial and bulk fields in the tactoids.

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Visualization and exploration of nematic liquid crystal (NLC) data is a challenging task due to the multidimensional and multivariate nature of the data. Simulation study of an NLC consists of multiple timesteps, where each timestep computes scalar, vector, and tensor parameters on a geometrical mesh. Scientists developing an understanding of liquid crystal interaction and physics require tools and techniques for effective exploration, visualization, and analysis of these data sets. Traditionally, scientists have used a combination of different tools and techniques like 2D plots, histograms, cut views, etc. for data visualization and analysis. However, such an environment does not provide the required insight into NLC datasets. This thesis addresses two areas of the study of NLC data---understanding of the tensor order field (the Q-tensor) and defect detection in this field. Tensor field understanding is enhanced by using a new glyph (NLCGlyph) based on a new design metric which is closely related to the underlying physical properties of an NLC, described using the Q-tensor. A new defect detection algorithm for 3D unstructured grids based on the orientation change of the director is developed. This method has been used successfully in detecting defects for both structured and unstructured models with varying grid complexity.

The main objective of this thesis is to obtain critical insight on the role of crystalline incompatibilities in strain and curvature, induced in presence of line defects i.e. dislocations and disclinations, on the energy and geometry of specific features of the local microstructure, and on the bulk mechanical response of nanocrystalline/ultra-fine grained materials. To that end, studies are performed at the (1) inter-atomic and fine scale, and (2) at the mesoscale. The modelling approach is based on the field dislocation and disclination mechanics theory of continuously representated dislocations and disclinations. New, thermodynamically rigorous, multi-scale elastic constitutive laws based on the couple stress theory are developed to capture the effect of strain and curvature incompatibilities on the Cauchy and couple stresses. A new meso-scale elasto-viscoplastic constitutive model of defect incompatibilities based on a fast Fourier transform technique is developed. The desired scale transitioning is achieved via novel phenomenological defect density transport equations and the newly developed elastic constitutive laws. At the fine scale, the model is applied to study energetic interactions between strain and curvature incompatibilities associated with grain boundaries and their influence on triple line energies. Results reveal that incompatible lattice strains have the most significant contribution to the energy. Incompatible lattice curvatures have negligible energetic contributions but are necessary to characterize the geometry of grain boundaries. Finally, both incompatible lattice strains and curvatures are necessary to capture the structure sensitive mechanical behavior of grain boundaries. At the mesoscale, deformation of nanocrystalline aggregates characterized by residual curvatures is studied to identify the impact of the latter's presence on the local and bulk mechanical response of the aggregate. Relaxation of local stresses generated from residual curvatures reproduces the effect of GB dislocation emission. Uniaxial tensile loading of nanocrystalline microstructures containing residual curvatures reveals a softening in the yield stress which could explain the breakdown in Hall-Petch law in the nanocrystalline regime. Next, the possibility of characterizing incompatibilities using X-ray or neutron diffraction techniques is tested. Results reveal that only strains and their gradients contribute to the broadening of diffraction peaks; curvatures and their gradients have no contribution. This study leads to the development of a new multi-scale averaged strain based Fourier technique for generating virtual diffraction peaks.

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Recent research shows that liquid crystals can be used to report the presence of different types of substances through optical amplication of ligand-receptor binding. In this work, simulations based on a coarse-grained method have been performed to study a class of liquid-crystal-based sensors. A tensor order parameter was used to model the liquid crystalline system and the Beris-Edwards formulation was employed to obtain the time evolution of a liquid crystal medium containing particles. The simulation cases are built using three-dimensional unstructured meshes and the simulation geometries studied include simple models involving spheres as well as detailed modeling for a protein. The dynamics of a liquid crystal medium confined between two solid walls has been studied in the presence of spherical particles and a representative biological macromolecule. Comparisons of steady state and transient solutions from the present study with corresponding results from molecular dynamics based simulations in the literature yield good agreements.

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