32 research outputs found
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
Effects of Friction and Disorder on the Quasi-Static Response of Granular Solids to a Localized Force
The response to a localized force provides a sensitive test for different
models of stress transmission in granular solids. The elasto-plastic models
traditionally used by engineers have been challenged by theoretical and
experimental results which suggest a wave-like (hyperbolic) propagation of the
stress, as opposed to the elliptic equations of static elasticity. Numerical
simulations of two-dimensional granular systems subject to a localized external
force are employed to examine the nature of stress transmission in these
systems as a function of the magnitude of the applied force, the frictional
parameters and the disorder (polydispersity). The results indicate that in
large systems (typically considered by engineers), the response is close to
that predicted by isotropic elasticity whereas the response of small systems
(or when sufficiently large forces are applied) is strongly anisotropic. In the
latter case the applied force induces changes in the contact network
accompanied by frictional sliding. The larger the coefficient of static
friction, the more extended is the range of forces for which the response is
elastic and the smaller the anisotropy. Increasing the degree of polydispersity
(for the range studied, up to 25%) decreases the range of elastic response.
This article is an extension of a previously published letter [1].Comment: 21 pages (PDFLaTeX), 24 figures (some of them bitmapped to save
space); submitted to Phys. Rev.
The problem of sharp notch in microstructured solids governed by dipolar gradient elasticity
In this paper, we deal with the asymptotic problem of a body of infinite extent with a notch (re-entrant corner) under remotely applied plane-strain or anti-plane shear loadings. The problem is formulated within the framework of the Toupin-Mindlin theory of dipolar gradient elasticity. This generalized continuum theory is appropriate to model the response of materials with microstructure. A linear version of the theory results by considering a linear isotropic expression for the strain-energy density that depends on strain-gradient terms, in addition to the standard strain terms appearing in classical elasticity. Through this formulation, a microstructural material constant is introduced, in addition to the standard Lamé constants . The faces of the notch are considered to be traction-free and a boundary-layer approach is followed. The boundary value problem is attacked with the asymptotic Knein-Williams technique. Our analysis leads to an eigenvalue problem, which, along with the restriction of a bounded strain energy, provides the asymptotic fields. The cases of a crack and a half-space are analyzed in detail as limit cases of the general notch (infinite wedge) problem. The results show significant departure from the predictions of the standard fracture mechanics