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Let K be the field of complex numbers. In this thesis we construct new examples of noncommutative surfaces of GK-dimension 4 using the language of formal and infinitesimal deformations as introduced by Gerstenhaber. Our approach is to find families of deformations of a certain well known GK-dimension 4 birationally commutative surface defined by Zhang and Smith in unpublished work cited in [YZ06], which we call A. Let B* and K* be respectively the bar and Koszul complexes of a PBW algebra C = KhV / (R) . We construct a graph whose vertices are elements of the free algebra KhV i and edges are relations in R. We define a map m2 : B2 ! K2 that extends to a chain map m* : B* → K*. This map allows the Gerstenhaber bracket structure to be transferred from the bar complex to the Koszul complex. In particular, m2 provides a mechanism for algorithmically determining the set of infinitesimal deformations with vanishing primary obstruction. Using the computer algebra package 'Sage' [Dev15] and a Python package developed by the author [Cam], we calculate the degree 2 component of the second Hochschild cohomology of A. Furthermore, using the map m2 we describe the variety U ⊆ HH2/2 (A) of infinitesimal deformations with vanishing primary obstruction. We further show that U decomposes as a union of 3 irreducible subvarieties Vg, Vq and Vu. More generally, let C be a Koszul algebra with relations R, and let E be a localisation of C at some (left and right) Ore set. Since R is homogeneous in degree two, there is an embedding R ,↪ C⊗C and in the following we identify R with its (nonzero) image under this map. We construct an injective linear map ~⋀ : HH²(C) → HH²(E) and prove that if f ∈ HH²(E) satisfies f(R) ⊆ C then f ∈ Im(~⋀). In this way we describe a relationship between infinitesimal deformations of C with those of E. Rogalski and Sierra [RS12] have previously examined a family of deformations of A arising from automorphism of the surface P1 X P1. By applying our understanding of the map ~⋀ we show that these deformations correspond to the variety of infinitesimal deformations Vg. Furthermore, we show that deformations defined similarly by automorphisms of other minimal rational surfaces also correspond to infinitesimal deformations lying in Vg. We introduce a new family of deformations of A, which we call Aq. We show that elements of this family have families of deformations arising from certain quantum analogues of geometric automorphisms of minimal rational surfaces, as defined by Alev and Dumas. Furthermore, we show that after taking the semi-classical limit q → 1 we obtain a family of deformations of A whose infinitesimal deformation lies in Vq. Finally, we apply a heuristic search method in the space of Hochschild 2-cocycles of A. This search yields another new family of deformations of A. We show that elements of this family are non-noetherian PBW noncommutative surfaces with GK-dimension 4. We further show that elements of this family can have as function skew field the division ring of the quantum plane Kq(u; v), the division ring of the first Weyl algebra D1(K) or the commutative field K(u; v).

Deformation of the orthodontic bracket upon wire engagement may result in torque dissipation within the bracket rather than transmission to the tooth and its supporting structures. The purpose of this study was: 1) quantify the amount of torque expression and 2) quantify any deformation as a result of such increasing torque expression. Digital image correlation is an accurate means of analyzing an orthodontic bracket's structural response to an applied moment of force created through archwire rotation in vitro. A sample of 30 Orthos and 30 OrthosTi brackets were tested with custom software using a novel application of digital image correlation. OrthosTi brackets produce significantly greater amounts of torque from 48 to 24 returning to a neutral position. Titanium brackets exhibit greater variance in torque expression but less variance in deformation. Orthos stainless steel brackets exhibit significantly greater amounts of bracket slot deformation in comparison to the OrthosTi. / Orthodontics

This study analyzes the thermo-mechanical response of a composite cylinder made up of two layers of linear isotropic viscoelastic materials that belong to the class of non-Thermorheologically Simple Material. The effect of time-varying temperature field due to unsteady heat conduction phenomenon is analyzed on the short term and long term material response in terms of stress, strain and displacement fields. The material properties of the two layers of the composite cylinder at any given location and time are assumed to depend on the temperature at that location at that given instant of time. Sequentially coupled analyses of heat conduction and deformation of viscoelastic composite cylinder is carried out to obtain the overall response. The stress and strain field developed in the composite cylinder is evaluated as the discontinuity in hoop stress and radial strain at the interface of the two layers caused due to mismatch in material properties may lead to delamination if it exceeds critical value. Analytical solution for the stress, strain and displacement fields of the viscoelastic composite cylinder is developed from the corresponding solution of linear elasticity problem by using the Correspondence Principle. The analytical solution for determining the temperature dependent stress, strain and displacement fields is further developed by incorporating the temperature dependence on the material properties and modeling the material as non-TSM. To analyze more complex geometry with general loading and boundary conditions, Finite Element(FE) analysis of the composite cylinder is performed and the results of analytical and FE method are found to be in good agreement. Parametric studies are carried out to understand the effect of change in material parameters namely the Prony coefficients in the transient creep compliance, characteristic of creep time in transient creep compliance and the instantaneous elastic compliance, on the overall response of the composite cylinder. The effect of different temperature dependent functions of the material properties, namely linear temperature variation and quadratic polynomial variation on the overall material response is also analyzed. It is observed that the effect of change in elastic properties significantly increases the jump in hoop stress and radial strain. It is also observed that when the materials are highly dependent on temperature the jump in radial strain and hoop stress increases significantly. The radial displacement also increases by a significant amount in both the cases.

It is well known that texture and microstructure play a crucial role in determining properties of metallic materials. The evolution of microstructure and texture during deformation and annealing of copper and some copper base alloys have been investigated to some extent. However, the knowledge about the role of the mode of deformation, particularly different variants of rolling deformation, is still very limited for deformation and annealing texture of two-phase copper alloys. Therefore it appeared important to study the influence of deformation path (in the present case, mode of rolling) on texture and microstructure in two-phase copper alloy Cu-40Zn alloy. Hot rolled bar of Cu-40Zn alloy (as-received material) was subjected to unidirectional and cross rolling ( multi-step cross-rolling) at room temperature with strain per pass (true strain) being constant for each step. In multi-step cross-rolling, the rolling direction was altered (rotated by 90°)after each pass. Strains corresponding to rolling reduction of 50% and 80% were given to as-received material through each of the different mode of rolling. In a second route, the as-received material was solution treated at 800°C for 4 hours first and then subjected to rolling in the same manner as above. A piece was cut from each of the as-rolled materials and was subjected to annealing at 560°C for one hour for recrystallization. The bulk textures were determined by measuring the pole figures at the center of the rolled as well as the annealed specimen using X-ray texture goniometer based on Schultz reflection geometry. Three dimensional texture analyses were carried out using the method of orientation distribution function(ODF). Micro-textures and associated microstructural parameters were determined using a Field Emission Gun Scanning Electron Microscope(FEG-SEM) operated at 20KV, equipped with Electron back scattering detector(EBSD). In the experimental material, texture was examined for both the α (fcc) and the β (ordered cubic) phases. In the present investigation, α phase of unidirectional rolled as-received material had Bs {011}<112> orientation as the strongest component whereas for multi-step cross rolled material P(BND) {011}<111> orientation had the maximum intensity, which could be obtained by rotating the Bs orientation and about ND.The texture development of β phase of as-received unidirectional rolled sample could be understood in terms of relaxed constraints Taylor model. The initial texture had a pronounced effect on texture development of α phase for solution treated alloy during deformation. This material exhibited very strong P(BND) {011}<111> orientation for unidirectional as well as for multi-step cross rolling. For cross rolled alloys, this orientation is promoted by two factors simultaneously, (i) initial texture and (ii) special attributes of cross rolling process. The volume fraction of cube oriented grains was very low for all recrystallized samples because of dominance of Bs orientation in the deformation texture plus formation of shear bands in the microstructure. The texture of β phase for unidirectionally rolled solution treated alloy got sharpened on annealing. However, strength of texture decreased with increasing deformation. Grain boundary (and CSL boundary) analyses were carried out with EBSD data. These analyses indicated that all the recrystallized samples had a high number of Σ3 boundaries. The proportion of Σ3 boundaries was higher in multi-step cross-rolled annealed material. The deformed material had higher number fraction of low angle boundary than any other special boundary. Solution treated material had an average grain size of α phase smaller than the as-received material. Another dimension of the present investigation was to characterize the microstructural features in three dimension(3D) in order to examine the morhphology of constituent phases using serial sectioning. In the present work, 3D studies were carried out on the alloy after post deformation annealing. The alignment of serial section images and generation of 3D image out of the stack of 2D images was carried out through standard software. The same was used to measure the suitable 3D microstructural parameters from the 2D sections. Three dimensional microstructural parameters like mean caliper diameter of β particle, number of β particles per unit volume ‘Nv’, surface to volume ratio for β phase particles (α- β interface) ‘Sv’, were calculated. Number of β particle intercepts per unit area ‘NA’ was determined by measuring number of β phase particle in each section. The volume of a β particle as calculated from the caliper diameter using three-dimensional microstructural analysis, which could not get directly determined with conventional two-dimensional microscopy.

The virtual worlds of Computer Graphics are populated by geometric objects, called models. Researchers have addressed the problem of synthesizing models automatically. Traditional modeling approaches often require a user to guide the synthesis process and to look after the geometry being synthesized, but user attention is expensive, and reducing user interaction is therefore desirable. I present a scheme for the automatic creation of geometry by deforming surfaces. My scheme includes a novel surface representation; it is an explicit representation consisting of points and edges, but it is not a traditional polygonal mesh. The novel surface representation is paired with a resampling policy to control the surface density and its evolution during deformation. The surface deforms with velocities assigned to its points through a set of deformation operators. Deformation operators avoid the manual computation and assignment of velocities, the operators allow a user to interactively assign velocities with minimal effort. Additionally, Petri nets are used to automatically deform a surface by mimicking a user assigning deformation operators. Furthermore, I present an algorithm to translate from the novel surface representations to a polygonal mesh. I demonstrate the utility of my model generation scheme with a gallery of models created automatically. The scheme's surface representation and resampling policy enables a surface to deform without requiring a user to control the deformation; self-intersections and hole creation are automatically prevented. The generated models show that my scheme is well suited to create organic-like models, whose surfaces have smooth transitions between surface features, but can also produce other kinds of models. My scheme allows a user to automatically generate varied instances of richly detailed models with minimal user interaction.

The virtual worlds of Computer Graphics are populated by geometric objects, called models. Researchers have addressed the problem of synthesizing models automatically. Traditional modeling approaches often require a user to guide the synthesis process and to look after the geometry being synthesized, but user attention is expensive, and reducing user interaction is therefore desirable. I present a scheme for the automatic creation of geometry by deforming surfaces. My scheme includes a novel surface representation; it is an explicit representation consisting of points and edges, but it is not a traditional polygonal mesh. The novel surface representation is paired with a resampling policy to control the surface density and its evolution during deformation. The surface deforms with velocities assigned to its points through a set of deformation operators. Deformation operators avoid the manual computation and assignment of velocities, the operators allow a user to interactively assign velocities with minimal effort. Additionally, Petri nets are used to automatically deform a surface by mimicking a user assigning deformation operators. Furthermore, I present an algorithm to translate from the novel surface representations to a polygonal mesh. I demonstrate the utility of my model generation scheme with a gallery of models created automatically. The scheme's surface representation and resampling policy enables a surface to deform without requiring a user to control the deformation; self-intersections and hole creation are automatically prevented. The generated models show that my scheme is well suited to create organic-like models, whose surfaces have smooth transitions between surface features, but can also produce other kinds of models. My scheme allows a user to automatically generate varied instances of richly detailed models with minimal user interaction.

This study analyzes the thermo-mechanical response of a composite cylinder made up of two layers of linear isotropic viscoelastic materials that belong to the class of non-Thermorheologically Simple Material. The effect of time-varying temperature field due to unsteady heat conduction phenomenon is analyzed on the short term and long term material response in terms of stress, strain and displacement fields. The material properties of the two layers of the composite cylinder at any given location and time are assumed to depend on the temperature at that location at that given instant of time. Sequentially coupled analyses of heat conduction and deformation of viscoelastic composite cylinder is carried out to obtain the overall response. The stress and strain field developed in the composite cylinder is evaluated as the discontinuity in hoop stress and radial strain at the interface of the two layers caused due to mismatch in material properties may lead to delamination if it exceeds critical value. Analytical solution for the stress, strain and displacement fields of the viscoelastic composite cylinder is developed from the corresponding solution of linear elasticity problem by using the Correspondence Principle. The analytical solution for determining the temperature dependent stress, strain and displacement fields is further developed by incorporating the temperature dependence on the material properties and modeling the material as non-TSM. To analyze more complex geometry with general loading and boundary conditions, Finite Element(FE) analysis of the composite cylinder is performed and the results of analytical and FE method are found to be in good agreement. Parametric studies are carried out to understand the effect of change in material parameters namely the Prony coefficients in the transient creep compliance, characteristic of creep time in transient creep compliance and the instantaneous elastic compliance, on the overall response of the composite cylinder. The effect of different temperature dependent functions of the material properties, namely linear temperature variation and quadratic polynomial variation on the overall material response is also analyzed. It is observed that the effect of change in elastic properties significantly increases the jump in hoop stress and radial strain. It is also observed that when the materials are highly dependent on temperature the jump in radial strain and hoop stress increases significantly. The radial displacement also increases by a significant amount in both the cases.

The mechanical properties of materials strongly depend on their microstructure. Therefore, engineering the material's microstructure can lead to improving its mechanical properties. One method for enhancing the strength of metallic materials consists of refining the grain size down to the nanometer scale. Such nanostructured materials possess remarkable strength without using conventional metallurgical strengthening methods. However, this strength often comes at the expense of workhardening capacity, thus favoring flow localization and loss of ductility and toughness. The deformation behavior of nanostructured metallic materials has been extensively studied in the literature. However, little is known of their fracture behavior. In this study, the mechanical behavior of a nanostructured, nearly pure material is investigated in order to link processing conditions, microstructure, and fracture locus in stress space. With focus laid on BCC materials which can undergo a ductile-to-brittle transition, Interstitial- Free (IF) steel is chosen. The microstructure is refined using Severe Plastic Deformation (SPD) to achieve ultra-fine grain (UFG) materials with grain sizes in the range 100nm- 1 mu m. Equal Channel Angular Extrusion (ECAE) is used to obtain three types of UFG-IF steel microstructures by varying the extrusion rate and processing temperature. The deformation behavior is investigated for the three UFG materials using round smooth bars and is compared with the behavior of the as-received material. The damage behavior and the fracture mechanisms are studied using tensile round notched bars with varying notch radii. The findings indicate a remarkable combination of strength and notch ductility at room temperature, including for the material with the finest microstructure. They also point to the need for careful characterization of temperature effects before such materials can be considered in structural applications.

The plastic deformation of stent during the implantation process, with considering the effect of artery and plaque, was investigated in this thesis. The stress-stretch relationships of porcine coronary arteries and aortas were obtained by the tensile test. The nonlinear ABAQUS finite element software was used in the analysis. The nonlinear-elastic, plastic and linear elastic material models for artery-plaque, stent and balloon were employed respectively. In this thesis, the initial folded balloon model was proposed to simulate the whole inflation and deflation process of balloon deformation. To investigate the effect of artery with plaque on the deformation of stent, the FEM model with considering the artery and plaque was proposed to simulate the expansion process of stents. The plaque¡¦s destruction during the expansion of stent was studied. The effects of plaque¡¦s thickness and the artery models on the expansion of stent were investigated. The effects of geometrical parameters and the influences on the deformation and the stress distribution of Palmaz type stent were discussed. The expansion ability and foreshortening of Chen and Tsai¡¦s stent were also assessed in this work. The results indicate that the limited expansion ability make Chen and Tsai¡¦s stent be not suitable for using as coronary stent. It might be used as the stent of carotid artery if its expansion ability improved by properly designing stent¡¦s diameter and geometry shape. The results reveal the foreshortening of Chen and Tsai¡¦s stent does not approach to zero as mentioned by Chen. A Chen modified and Tsai¡¦s stent was also proposed in this study. The simulation results show that the foreshortening in the Chen modified type is improved.