Topic
Distribution (differential geometry)
About: Distribution (differential geometry) is a research topic. Over the lifetime, 911 publications have been published within this topic receiving 10149 citations.
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TL;DR: The geometrical formulation of continuum mechanics provides a powerful approach to understand and solve problems in anelasticity where an elastic deformation is combined with a non-elastic component arising from defects, thermal stresses, growth effects or other effects leading to residual stresses.
Abstract: The geometrical formulation of continuum mechanics provides us with a powerful approach to understand and solve problems in anelasticity where an elastic deformation is combined with a non-elastic component arising from defects, thermal stresses, growth effects or other effects leading to residual stresses. The central idea is to assume that the material manifold, prescribing the reference configuration for a body, has an intrinsic, non-Euclidean, geometrical structure. Residual stresses then naturally arise when this configuration is mapped into Euclidean space. Here, we consider the problem of discombinations (a new term that we introduce in this paper), that is, a combined distribution of fields of dislocations, disclinations and point defects. Given a discombination, we compute the geometrical characteristics of the material manifold (curvature, torsion, non-metricity), its Cartan's moving frames and structural equations. This identification provides a powerful algorithm to solve semi-inverse problems with non-elastic components. As an example, we calculate the residual stress field of a cylindrically symmetric distribution of discombinations in an infinite circular cylindrical bar made of an incompressible hyperelastic isotropic elastic solid.
31 citations
TL;DR: In this paper, the full Lorentzian Einstein equation is reduced to equations on a family of Einstein Riemannian metrics, and the general Walker coordinates can be simplified.
Abstract: We study transformations of coordinates on a Lorentzian Einstein manifold with a parallel distribution of null lines and show that the general Walker coordinates can be simplified. In these coordinates, the full Lorentzian Einstein equation is reduced to equations on a family of Einstein Riemannian metrics.
31 citations
TL;DR: In this article, the fundamental theory of accreting bodies for finite deformations is explained using the concept of the bundle of a differentiable manifold that enables one to construct a clear classification of the accretion processes.
31 citations
TL;DR: In this paper, a Riemannian manifold whose geodesics trace the edges of the mesh is defined, and a function @f, proportional to the logarithm of the cell size, is shown to obey the Poisson equation, with localized charges corresponding to irregular vertices.
30 citations
TL;DR: In this article, the authors determine the asymptotics of the joint eigenfunctions of the torus action on a toric Kahler variety and show that they behave like Gaussians centered at the corresponding classical torus, and that there is a universal Gaussian scaling limit of the distribution function near its center.
Abstract: We determine the asymptotics of the joint eigenfunctions of the torus action on a toric Kahler variety. Such varieties are models of completely integrable systems in complex geometry. We first determine the pointwise asymptotics of the eigenfunctions, which show that they behave like Gaussians centered at the corresponding classical torus. We then show that there is a universal Gaussian scaling limit of the distribution function near its center. We also determine the limit distribution for the tails of the eigenfunctions on large length scales. These are not universal but depend on the global geometry of the toric variety and in particular on the details of the exponential decay of the eigenfunctions away from the classically allowed set.
30 citations