Browsing by Subject "Calculus of Variations"

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    Asymptotic models in magnetostriction with application to design of sensors.
    (2012-04) Krishnan, Shankar Narayan
    Magnetostrictive wires of diameter in the nanometer scale have been proposed for application as acoustic sensors [Downey et al., 2008], [Yang et al., 2006]. The sensing mechanism is expected to operate in the bending regime. In the first part of this work, we derive a variational theory for the bending of magnetostrictive nanowires starting from a full 3-dimensional continuum theory of magnetostriction. We recover a theory which looks like a typical Euler-Bernoulli bending model but includes an extra term contributed by the magnetic part of the energy. The solution of this variational theory for an important, newly developed magnetostricitve alloy called Galfenol ¡ cf. [Clark et al., 2000] ¢ is compared with the result of experiments on actual nanowires ¡ cf. [Downey, 2008] ¢ which shows agreement. In the next part of this thesis, Multilayered wires of diameter in the nanometer scale with periodic layering of non-magnetic copper and ferromagnetic galfenol segments are studied. The numerical computation of the physics of magnetization for such geometries is very costly computationally. We use the theory of periodic homogenization to understand the overall behavior of such structures. We first determine a “homogenized theory” after which this “homogenized model” is used to study the nucleation and stability of staturated states. Thus we get a broad generalization of what is known in the magnetic literature as the “fanning model” first introduced in [Jacobs and Bean, 1955] for a chain of spheres geometry. Some further numerical work on computing M vs H curves for such geometries is also presented.
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    Topics in the mathematical theory of nonlinear elasticity.
    (2012-08) Li, Hui
    My thesis consists of two interrelated parts. The first part lies in the field of the mathematical theory of nonlinear elasticity, and it concerns the rigorous derivation of theories for elastic shells. The second part concerns modeling and analyzing of shells with residual stresses. The approach for both parts is based on the refined methods in Calculus of Variations (notably the so-called $\Gamma$-convergence) and a combination of the arguments in modern Mathematical Analysis and Riemannian Geometry. More precisely, in chapter 2-3, we derive the von K\'arm\'an theory for variable thickness shells and also the von K\'arm\'an theory for incompressible shells with uniform thickness. In chapter 4-5, we first establish the Kirchhoff theory for non-Euclidean shells and its incompressible counterpart. Then, we also derive the incompatible F\"oppl-von K\'arm\'an theory for prestrained shells with variable thickness, calculate the associated Euler-Lagrange equations and found the convergence of equilibria. Finally, the incompatible F\"oppl-von K\'arm\'an theory for incompressible prestrained shells and the associated Euler-Lagrange equations are investigated.