Mathematical models of martensitic microstructure

Martensitic microstructures are studied using variational models based on nonlinear elasticity. Some relevant mathematical tools from nonlinear analysis are described, and applications given to austenite-martensite interfaces and related topics

Nematic liquid crystals : from Maier-Saupe to a continuum theory

We define a continuum energy functional in terms of the mean-field Maier-Saupe free energy, that describes both spatially homogeneous and inhomogeneous systems. The Maier-Saupe theory defines the main macroscopic variable, the Q-tensor order parameter, in terms of the second moment of a probability distribution function. This definition requires the eigenvalues of Q to be bounded both from below and above. We define a thermotropic bulk potential which blows up whenever the eigenvalues tend to these lower and upper bounds. This is in contrast to the Landau-de Gennes theory which has no such penalization. We study the asymptotics of this bulk potential in different regimes and discuss phase transitions predicted by this model

Controllability and stabilizability of distributed bilinear systems: Recent results and open problems

This paper describes recent results for controlling and stabilizing control systems of the form ú(t) = Au(t) + p(t) B(u(t)) where A is the infinitesimal generator C∞ semigroup on a Banach space X, B' map from X into X, and p(t) is a real valued control. Application to a vibrating beam problem is given for illusstration of the theory

Controllability for Distributed Bilinear Systems

This paper studies controllability of systems of the form ${{dw} / {dt}} = \mathcal {A}w + p(t)\mathcal {B}w$ where $\mathcal{A}$ is the infinitesimal generator of a $C^0$ semigroup of bounded linear operators $e^{\mathcal{A}t}$ on a Banach space $X$, $\mathcal{B}:X \to X$ is a $C^1$ map, and $p \in L^1 ([0,T];\mathbb{R})$ is a control. The paper (i) gives conditions for elements of $X$ to be accessible from a given initial state $w_0$ and (ii) shows that controllability to a full neighborhood in $X$ of $w_0$ is impossible for $\dim X = \infty$. Examples of hyperbolic partial differential equations are provided

Nano-structures at martensite macrotwin interfaces in Ni65Al35Ni_{65}Al_{35}

The atomic configurations at macrotwin interfaces between microtwinned martensite plates in $Ni_{65}Al_{35}$ material are investigated using transmission electron microscopy. The observed structures are interpreted in view of possible formation mechanisms for these interfaces. A distinction is made between cases in which the microtwins, originating from mutually perpendicular \{110\} austenite planes, enclose a final angle larger or smaller than $90^{\circ}$. Two different configurations, a crossing and a step type are described. Depending on the actual case, tapering, bending and tip splitting of the smaller microtwin variants are observed. The most reproducible deformations occur in a region of approximately 5-10nm width around the interface while a variety of structural defects are observed further away from the interface. These structures and deformations are interpreted in terms of the coalescence of two separately nucleated microtwinned martensite plates and the need to accommodate remaining stresses

Chiral-symmetry breaking in dual QCD

In the context of the formulation of QCD with dual potentials, we show that chiral-symmetry breaking occurs only in the confined state. Therefore, the transition temperature, beyond which chiral symmetry is restored, is the same as the deconfinement temperature. To carry out the calculation, it is necessary to couple quarks to dual gluons. We indicate how this is done (to lowest order in the magnetic coupling constant) and give the Feynman rules for quark–dual-gluon vertices

A Constituent Quark Anti-Quark Effective Lagrangian Based on the Dual Superconducting Model of Long Distance QCD

We review the assumptions leading to the description of long distance QCD by a Lagrangian density expressed in terms of dual potentials. We find the color field distribution surrounding a quark anti-quark pair to first order in their velocities. Using these distributions we eliminate the dual potentials from the Lagrangian density and obtain an effective interaction Lagrangian $L_I ( \vec x_1 \, , \vec x_2 \, ; \vec v_1 \, , \vec v_2 )$ depending only upon the quark and anti-quark coordinates and velocities, valid to second order in their velocities. We propose $L_I$ as the Lagrangian describing the long distance interaction between constituent quarks. Elsewhere we have determined the two free parameters in $L_I$, $\alpha_s$ and the string tension $\sigma$, by fitting the 17 known levels of $b \bar b$ and $c \bar c$ systems. Here we use $L_I$ at the classical level to calculate the leading Regge trajectory. We obtain a trajectory which becomes linear at large $M^2$ with a slope $\alpha' \simeq .74 \, \hbox{GeV}^{-1}$, and for small $M^2$ the trajectory bends so that there are no tachyons. For a constituent quark mass between 100 and 150 MeV this trajectory passes through the two known Regge recurrences of the $\pi$ meson. In this paper, for simplicity of presentation, we have treated the quarks as spin-zero particles.Comment: {\bf 32,UW/PT94-0

Static quark potential according to the dual-superconductor picture of QCD

We use the effective action describing long-range QCD, which predicts that QCD behaves as a dual superconductor, to derive the interaction energy between two heavy quarks as a function of separation. The dual-superconductor field equations are solved in an approximation in which the boundary between the superconducting vacuum and the region of normal vacuum surrounding the quarks is sharp. Further, non-Abelian effects are neglected. The resulting heavy-quark potential is linear in separation at large separation, and Coulomb-like at small separation. Overall it agrees very well with phenomenologically determined potentials

Quantized electric-flux-tube solutions to Yang-Mills theory

We suggest that long-distance Yang-Mills theory is more conveniently described in terms of electric rather than the customary magnetic vector potentials. On this basis we propose as an effective Lagrangian for this regime the most simple gauge-invariant (under the magnetic rather than electric gauge group) and Lorentz-invariant Lagrangian which yields a 1/q^4 gluon propagator in the Abelian limit. The resulting classical equations of motion have solutions corresponding to tubes of color electric flux quantized in units of e/2 (e is the Yang-Mills coupling constant). To exponential accuracy the electric color energy is contained in a cylinder of finite radius, showing that continuum Yang-Mills theory has excitations which are confined tubes of color electric flux. This is the criterion for electric confinement of color