81 research outputs found
Line and Point Defects in Nonlinear Anisotropic Solids
In this paper, we present some analytical solutions for the stress fields of
nonlinear anisotropic solids with distributed line and point defects. In
particular, we determine the stress fields of i) a parallel
cylindrically-symmetric distribution of screw dislocations in infinite
orthotropic and monoclinic media, ii) a cylindrically-symmetric distribution of
parallel wedge disclinations in an infinite orthotropic medium, iii) a
distribution of edge dislocations in an orthotropic medium, and iv) a
spherically-symmetric distribution of point defects in a transversely isotropic
spherical ball
Structure of Defective Crystals at Finite Temperatures: A Quasi-Harmonic Lattice Dynamics Approach
In this paper we extend the classical method of lattice dynamics to defective
crystals with partial symmetries. We start by a nominal defect configuration
and first relax it statically. Having the static equilibrium configuration, we
use a quasiharmonic lattice dynamics approach to approximate the free energy.
Finally, the defect structure at a finite temperature is obtained by minimizing
the approximate Helmholtz free energy. For higher temperatures we take the
relaxed configuration at a lower temperature as the reference configuration.
This method can be used to semi-analytically study the structure of defects at
low but non-zero temperatures, where molecular dynamics cannot be used. As an
example, we obtain the finite temperature structure of two 180^o domain walls
in a 2-D lattice of interacting dipoles. We dynamically relax both the position
and polarization vectors. In particular, we show that increasing temperature
the domain wall thicknesses increase
Estimating Terminal Velocity of Rough Cracks in the Framework of Discrete Fractal Fracture Mechanics
In this paper we first obtain the order of stress singularity for a
dynamically propagating self-affine fractal crack. We then show that there is
always an upper bound to roughness, i.e. a propagating fractal crack reaches a
terminal roughness. We then study the phenomenon of reaching a terminal
velocity. Assuming that propagation of a fractal crack is discrete, we predict
its terminal velocity using an asymptotic energy balance argument. In
particular, we show that the limiting crack speed is a material-dependent
fraction of the corresponding Rayleigh wave speed
Geometric Phases of Nonlinear Planar -Pendula via Cartan's Moving Frames
We study the geometric phases of nonlinear planar -pendula with continuous
rotational symmetry. In the Hamiltonian framework, the geometric structure of
the phase space is a principal fiber bundle, i.e., a base, or shape manifold
, and fibers along the symmetry direction attached
to it. The symplectic structure of the Hamiltonian dynamics determines the
connection and curvature forms of the shape manifold. Using Cartan's structural
equations with zero torsion we find an intrinsic (pseudo) Riemannian metric for
the shape manifold. For a double pendulum the metric is pseudo-Riemannian if
the total angular momentum , and the shape manifold is an
expanding spacetime with the Robertson-Walker metric and positive curvature.
For , the shape manifold is the hyperbolic plane
with negative curvature. We then generalize our results to free -pendula. We
show that the associated shape manifold~ is reducible to the
product manifold of hyperbolic planes ~(),
or Robertson-Walker D spacetimes~(). We then consider
-pendula subject to time-dependent self-equilibrated moments. The extended
autonomous Hamiltonian system is considered. The associated shape manifold is a
product space of hyperbolic planes ~(), or
Robertson-Walker D spacetimes~(). In either case, the
geometric phase follows by integrating a -form given by the sum of the
sectional curvature forms of . The Riemannian structure of the
shape manifold provides an intrinsic measure of the closeness of one shape to
another in terms of curvature, or induced geometric phase
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