Thesis (PhD)--Montana State University--Bozeman, 2009. / Typescript. Chairperson, Graduate Committee: Maurice J. Burke. Includes bibliographical references (leaves 179-186).

Thesis (D.A.)--Illinois State University, 1994. / Title from title page screen, viewed March 31, 2006. Dissertation Committee: Robert G. Hathway (chair), Lynn H. Brown, John A. Dossey, Arnold J. Insel, Patricia H. Klass. Includes bibliographical references (leaves 33-34) and abstract. Also available in print.

Thesis (Ph. D.)--Rutgers University, 2009. / "Graduate Program in Mathematics." Includes bibliographical references (p. 151-154).

The application of techniques from nonlinear analysis to materials science has seen great developments in the recent years and it has really been a driving force for substantial mathematical research in the area of partial differential equations and the multi-dimensional calculus of variations. This thesis has been motivated by two recent and remarkable experimental observations of H. Seiner in shape-memory alloys which we attempt to interpret mathematically. Much of the work is original and has given rise to deep problems in the calculus of variations. Firstly, we study the formation of non-classical austenite-martensite interfaces. Ball & Carstensen (1997, 1999) theoretically investigated the possibility of the occurrence of such interfaces and studied the cubic-to-tetragonal case extensively. In this thesis, we present an analysis of non-classical austenite-martensite interfaces recently observed by Seiner et al.~in a single crystal of a CuAlNi shape-memory alloy, undergoing a cubic-to-orthorhombic transition. We show that these can be described by the general nonlinear elasticity model and we make some predictions regarding the admissible volume fractions of the martensitic variants involved, as well as the habit plane normals. Interestingly, in the above experimental observations, the interface between the austenite and the martensitic configuration is never exactly planar, but rather slightly curved, resulting from the pattern of martensite not being exactly homogeneous. However, it is not clear how one can reconstruct the inhomogeneous configuration as a stress-free microstructure and, instead, a theoretical approach is followed. In this approach, a general method is provided for the construction of a compatible curved austenite-martensite interface and, by exploiting the structure of quasiconvex hulls, the existence of curved interfaces is shown in two and three dimensions. As far as the author is aware of, this is the first construction of such a curved austenite-martensite interface. Secondly, we study the nucleation of austenite in a single crystal of a CuAlNi shape-memory alloy consisting of a single variant of stabilized 2H martensite. The nucleation process is induced by localized heating and it is observed that, regardless of where the localized heating is applied, the nucleation points are always located at one of the corners of the sample - a rectangular parallelepiped in the austenite. Using a simplified nonlinear elasticity model, we propose an explanation for the location of the nucleation points by showing that the martensite is a local minimizer of the energy with respect to localized variations in the interior, on faces and edges of the sample, but not at some corners, where a localized microstructure can lower the energy. The result for the interior, faces and edges is established by showing that the free-energy function satisfies a set of quasiconvexity conditions at the stabilized variant throughout the specimen, provided this is suitably cut. The proofs of quasiconvexity are based on a rigidity argument and are specific to the change of symmetry in the phase transformation. To the best of the author's knowledge, quasiconvexity conditions at edges and corners have not been considered before.

Disputats, Göteborg : Chalmers Tekniska Högskola ; Göteborgs universitet, 2008. / Findes også på internet. Med litteraturhenvisninger.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1970. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

No description available.

This thesis is concerned with mathematical modelling and optimal control of constrained systems. Each of the systems under consideration is a system that can be controlled by one of the variables, and this control is subject to constraints. First, we consider middle-distance running where a runner's horizontal propulsive force is the control which is constrained to be within a given range. Middle-distance running is typically a strategy-intensive race as slipstreaming effects come into play since speeds are still relatively fast and runners can leave their starting lane. We formulate a two-runner coupled model and determine optimal strategies using optimal control theory. Second, we consider two applications of control systems with delay related to R&D expenditure. The first of these applications relates to the defence industry. The second relates to the pharmaceutical industry. Both applications are characterised by a long delay between initial investment in R&D and seeing the benefits of R&D realised. We formulate models tailored to each application and use optimal control theory to determine the optimal proportion of available funds to invest in R&D over a given time horizon. Third, we consider a mathematical model of urban burglary based on the Short model. We make some modifications to this model including the addition of deterrence due to police officer presence. Police officer density is the control variable, which is constrained due to a finite number of police officers. We look at different control strategies for the police and their effect on burglary hot-spot formation.

This thesis is devoted to the mathematical analysis of models describing the energy of defects in crystalline solids via variational methods. The first part of this work studies a discrete model describing the energy of a point defect in a one dimensional chain of atoms. We derive an expansion of the ground state energy using Gamma-convergence, following previous work on similar models [BDMG99,BC07,SSZ11]. The main novelty here is an explicit characterisation of the first order limit as the solution of a variational problem in an infinite lattice. Analysing this variational problem, we prove a regularity result for the perturbation caused by the defect, and demonstrate the order of the next term in the expansion. The second main topic is a discrete model describing screw dislocations in body centred cubic crystals. We formulate an anti plane lattice model which describes the energy difference between deformations and, using the framework defined in [AO05], provide a kinematic description of the Burgers vector, which is a key geometric quantity used to describe dislocations. Apart from the anti plane restriction, this model is invariant under all the natural symmetries of the lattice and in particular allows for the creation and annihilation of dislocations. The energy difference formulation enables us to provide a clear definition of what it means to be a stable deformation. The main results of the analysis of this model are then first, a proof that deformations with unit net Burgers vector exist as globally stable states in an infinite body, and second, that deformations containing multiple screw dislocations exist as locally stable states in both infinite bodies and finite convex bodies. To prove the former result, we establish coercivity with respect to the elastic strain, and exploit a concentration compactness principle. In the latter case, we use a form of the inverse function theorem, proving careful estimates on the residual and stability of an ansatz which combines continuum linear elasticity theory with an atomistic core correction.

The thesis concerns the mathematical study of the calculus of variations and its application to liquid crystals. In the first chapter we examine vectorial problems in the calculus of variations with an additional pointwise constraint so that any admissible function <strong>n</strong> ε W^1,1(ΩM), and M is a manifold of suitable regularity. We formulate necessary and sufficient conditions for any given state <strong>n</strong> to be a strong or weak local minimiser of I. This is achieved using a nearest point projection mapping in order to use the more classical results which apply in the absence of a constraint. In the subsequent chapters we study various static continuum theories of liquid crystals. More specifically we look to explain a particular cholesteric fingerprint pattern observed by HP Labs. We begin in Chapter 2 by focusing on a specific cholesteric liquid crystal problem using the theory originally derived by Oseen and Frank. We find the global minimisers for general elastic constants amongst admissible functions which only depend on a single variable. Using the one-constant approximation for the Oseen-Frank free energy, we then show that these states are global minimisers of the three-dimensional problem if the pitch of the cholesteric liquid crystal is sufficiently long. Chapter 3 concerns the application of the results from the first chapter to the situations investigated in the second. The local stability of the one-dimensional states are quantified, analytically and numerically, and in doing so we unearth potential shortcomings of the classical Oseen-Frank theory. In Chapter 4, we ascertain some equivalence results between the continuum theories of Oseen and Frank, Ericksen, and Landau and de Gennes. We do so by proving lifting results, building on the work of Ball and Zarnescu, which relate the regularity of line and vector fields. The results prove to be interesting as they show that for a director theory to respect the head to tail symmetry of the liquid crystal molecules, the appropriate function space for the director field is S BV² (Ω,S<sup>2,/sup>). We take this idea and in the final chapter we propose a mathematical model of liquid crystals based upon the Oseen-Frank free energy but using special functions of bounded variation. We establish the existence of a minimiser, forms of the Euler-Lagrange equation, and find solutions of the Euler-Lagrange equation in some simple cases. Finally we use our proposed model to re-examine the same problems from Chapter 2. By doing so we extend the analysis we were able to achieve using Sobolev spaces and predict the existence of multi-dimensional minimisers consistent with the known experimental properties of high-chirality cholesteric liquid crystals.