The magnetic flux dynamics of type-II superconductors within the critical
state regime is posed in a generalized framework, by using a variational theory
supported by well established physical principles. The equivalence between the
variational statement and more conventional treatments, based on the solution
of the differential Maxwell equations together with appropriate conductivity
laws is shown. Advantages of the variational method are emphasized, focusing on
its numerical performance, that allows to explore new physical scenarios. In
particular, we present the extension of the so-called double critical state
model to three dimensional configurations in which only flux transport
(T-states), cutting (C-states) or both mechanisms (CT-states) occur. The theory
is applied to several problems. First, we show the features of the transition
from T to CT states. Second, we give a generalized expression for the flux
cutting threshold in 3-D and show its relevance in the slab geometry. In
addition, several models that allow to treat flux depinning and cutting
mechanisms are compared. Finally, the longitudinal transport problem (current
is applied parallel to the external magnetic field) is analyzed both under T
and CT conditions. The complex interaction between shielding and transport is
solved.Comment: 21 figures, submitted for publicatio
Critical state problems which incorporate more than one component for the
magnetization vector of hard superconductors are investigated. The theory is
based on the minimization of a cost functional ${\cal C}[\vec{H}(\vec{x})]$
which weighs the changes of the magnetic field vector within the sample. We
show that Bean's simplest prescription of choosing the correct sign for the
critical current density $J_c$ in one dimensional problems is just a particular
case of finding the components of the vector $\vec{J}_c$. $\vec{J}_c$ is
determined by minimizing ${\cal C}$ under the constraint $\vec{J}\in\Delta
(\vec{H},\vec{x})$, with $\Delta$ a bounded set. Upon the selection of
different sets $\Delta$ we discuss existing crossed field measurements and
predict new observable features. It is shown that a complex behavior in the
magnetization curves may be controlled by a single external parameter, i.e.:
the maximum value of the applied magnetic field $H_m$.Comment: 10 pages, 9 figures, accepted in Phys. Rev.
The Rusk-Skinner formalism was developed in order to give a geometrical unified formalism for describing mechanical systems. It incorporates all the characteristics of Lagrangian and Hamiltonian descriptions of these systems ͑including dynamical equations and solutions, constraints, Legendre map, evolution operators, equivalence, etc.͒. In this work we extend this unified framework to first-order classical field theories, and show how this description comprises the main features of the Lagrangian and Hamiltonian formalisms, both for the regular and singular cases. This formulation is a first step toward further applications in optimal control theory for partial differential equations.
Coarse-grained flux density profiles in type-II superconductors with nonparallel vortex configurations are obtained by a proposed phenomenological least action principle. We introduce a functional C[ H] which is minimized under a constraint of the kind J ∈ ∆( H, x), where ∆ is a bounded set.In particular, we choose the isotropic case | J | ≤ J c (H), for which the field penetration profiles H( x, t) are derived when a changing external excitation is applied. Faraday's law, and the principle of minimum entropy production rate for stationary thermodynamic processes dictate the evolution of the system.Calculations based on the model can reproduce the physical phenomena of flux transport and consumption, and the striking effect of magnetization collapse in crossed field measurements. PACS number(s): 41.20. Gz,74.60.Jg, 74.60.Ge, 02.30.Xx Typeset using REVT E X 1
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