1,047 research outputs found
An elastoplastic theory of dislocations as a physical field theory with torsion
We consider a static theory of dislocations with moment stress in an
anisotropic or isotropic elastoplastical material as a T(3)-gauge theory. We
obtain Yang-Mills type field equations which express the force and the moment
equilibrium. Additionally, we discuss several constitutive laws between the
dislocation density and the moment stress. For a straight screw dislocation, we
find the stress field which is modified near the dislocation core due to the
appearance of moment stress. For the first time, we calculate the localized
moment stress, the Nye tensor, the elastoplastic energy and the modified
Peach-Koehler force of a screw dislocation in this framework. Moreover, we
discuss the straightforward analogy between a screw dislocation and a magnetic
vortex. The dislocation theory in solids is also considered as a
three-dimensional effective theory of gravity.Comment: 38 pages, 6 figures, RevTe
Stress-free states of continuum dislocation fields: Rotations, grain boundaries, and the Nye dislocation density tensor
We derive general relations between grain boundaries, rotational
deformations, and stress-free states for the mesoscale continuum Nye
dislocation density tensor. Dislocations generally are associated with
long-range stress fields. We provide the general form for dislocation density
fields whose stress fields vanish. We explain that a grain boundary (a
dislocation wall satisfying Frank's formula) has vanishing stress in the
continuum limit. We show that the general stress-free state can be written
explicitly as a (perhaps continuous) superposition of flat Frank walls. We show
that the stress-free states are also naturally interpreted as configurations
generated by a general spatially-dependent rotational deformation. Finally, we
propose a least-squares definition for the spatially-dependent rotation field
of a general (stressful) dislocation density field.Comment: 9 pages, 3 figure
State-constrained control-affine parabolic problems II: Second order sufficient optimality conditions
In this paper we consider an optimal control problem governed by a semilinear heat equation with bilinear control-state terms and subject to control and state constraints. The state constraints are of integral type, the integral being with respect to the space variable. The control is multidimensional. The cost functional is of a tracking type and contains a linear term in the control variables. We derive second order sufficient conditions relying on the Goh transform
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State-constrained control-affine parabolic problems I: First and second order necessary optimality conditions
In this paper we consider an optimal control problem governed by a semilinear heat equation with bilinear control-state terms and subject to control and state constraints. The state constraints are of integral type, the integral being with respect to the space variable. The control is multidimensional. The cost functional is of a tracking type and contains a linear term in the control variables. We derive second order necessary conditions relying on the concept of alternative costates and quasi-radial critical directions
Cartan's spiral staircase in physics and, in particular, in the gauge theory of dislocations
In 1922, Cartan introduced in differential geometry, besides the Riemannian
curvature, the new concept of torsion. He visualized a homogeneous and
isotropic distribution of torsion in three dimensions (3d) by the "helical
staircase", which he constructed by starting from a 3d Euclidean space and by
defining a new connection via helical motions. We describe this geometric
procedure in detail and define the corresponding connection and the torsion.
The interdisciplinary nature of this subject is already evident from Cartan's
discussion, since he argued - but never proved - that the helical staircase
should correspond to a continuum with constant pressure and constant internal
torque. We discuss where in physics the helical staircase is realized: (i) In
the continuum mechanics of Cosserat media, (ii) in (fairly speculative) 3d
theories of gravity, namely a) in 3d Einstein-Cartan gravity - this is Cartan's
case of constant pressure and constant intrinsic torque - and b) in 3d Poincare
gauge theory with the Mielke-Baekler Lagrangian, and, eventually, (iii) in the
gauge field theory of dislocations of Lazar et al., as we prove for the first
time by arranging a suitable distribution of screw dislocations. Our main
emphasis is on the discussion of dislocation field theory.Comment: 31 pages, 8 figure
Mesoscale theory of grains and cells: crystal plasticity and coarsening
Solids with spatial variations in the crystalline axes naturally evolve into
cells or grains separated by sharp walls. Such variations are mathematically
described using the Nye dislocation density tensor. At high temperatures,
polycrystalline grains form from the melt and coarsen with time: the
dislocations can both climb and glide. At low temperatures under shear the
dislocations (which allow only glide) form into cell structures. While both the
microscopic laws of dislocation motion and the macroscopic laws of coarsening
and plastic deformation are well studied, we hitherto have had no simple,
continuum explanation for the evolution of dislocations into sharp walls. We
present here a mesoscale theory of dislocation motion. It provides a
quantitative description of deformation and rotation, grounded in a microscopic
order parameter field exhibiting the topologically conserved quantities. The
topological current of the Nye dislocation density tensor is derived from a
microscopic theory of glide driven by Peach-Koehler forces between dislocations
using a simple closure approximation. The resulting theory is shown to form
sharp dislocation walls in finite time, both with and without dislocation
climb.Comment: 5 pages, 3 figure
Aharonov-Bohm Effect and Disclinations in an Elastic Medium
In this work we investigate quasiparticles in the background of defects in
solids using the geometric theory of defects. We use the parallel transport
matrix to study the Aharonov-Bohm effect in this background. For quasiparticles
moving in this effective medium we demonstrate an effect similar to the
gravitational Aharonov- Bohm effect. We analyze this effect in an elastic
medium with one and defects.Comment: 6 pages, Revtex
On the incompatibility of strains and its application to mesoscopic studies of plasticity
Structural transitions are invariably affected by lattice distortions. If the
body is to remain crack-free, the strain field cannot be arbitrary but has to
satisfy the Saint-Venant compatibility constraint. Equivalently, an
incompatibility constraint consistent with the actual dislocation network has
to be satisfied in media with dislocations. This constraint can be incorporated
into strain-based free energy functionals to study the influence of
dislocations on phase stability. We provide a systematic analysis of this
constraint in three dimensions and show how three incompatibility equations
accommodate an arbitrary dislocation density. This approach allows the internal
stress field to be calculated for an anisotropic material with spatially
inhomogeneous microstructure and distribution of dislocations by minimizing the
free energy. This is illustrated by calculating the stress field of an edge
dislocation and comparing it with that of an edge dislocation in an infinite
isotropic medium. We outline how this procedure can be utilized to study the
interaction of plasticity with polarization and magnetization.Comment: 6 pages, 2 figures; will appear in Phys. Rev.
Riemann-Cartan Space-times of G\"odel Type
A class of Riemann-Cartan G\"odel-type space-times are examined in the light
of the equivalence problem techniques. The conditions for local space-time
homogeneity are derived, generalizing previous works on Riemannian G\"odel-type
space-times. The equivalence of Riemann-Cartan G\"odel-type space-times of this
class is studied. It is shown that they admit a five-dimensional group of
affine-isometries and are characterized by three essential parameters : identical triads () correspond to locally
equivalent manifolds. The algebraic types of the irreducible parts of the
curvature and torsion tensors are also presented.Comment: 24 pages, LaTeX fil
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