Approximations of the Wiener sausage and its curvature measures

Jan Rataj; Evgeny Spodarev; Daniel Meschenmoser

doi:10.1214/09-aap596

Approximations of the Wiener sausage and its curvature measures

The Annals of Applied Probability ◽

10.1214/09-aap596 ◽

2009 ◽

Vol 19 (5) ◽

pp. 1840-1859 ◽

Cited By ~ 1

Author(s):

Jan Rataj ◽

Evgeny Spodarev ◽

Daniel Meschenmoser

Keyword(s):

Wiener Sausage ◽

Curvature Measures

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A new approach to curvature measures in linear shell theories

Mathematics and Mechanics of Solids ◽

10.1177/1081286520972752 ◽

2020 ◽

pp. 108128652097275

Author(s):

Miroslav Šilhavý

Keyword(s):

Strain Tensor ◽

Middle Surface ◽

Surface Strain ◽

Affine Function ◽

Shear Deformations ◽

Strain Measure ◽

Bending Strain ◽

Curvature Measures ◽

Tensor Formula ◽

Strain Measures

The paper presents a coordinate-free analysis of deformation measures for shells modeled as 2D surfaces. These measures are represented by second-order tensors. As is well-known, two types are needed in general: the surface strain measure (deformations in tangential directions), and the bending strain measure (warping). Our approach first determines the 3D strain tensor E of a shear deformation of a 3D shell-like body and then linearizes E in two smallness parameters: the displacement and the distance of a point from the middle surface. The linearized expression is an affine function of the signed distance from the middle surface: the absolute term is the surface strain measure and the coefficient of the linear term is the bending strain measure. The main result of the paper determines these two tensors explicitly for general shear deformations and for the subcase of Kirchhoff-Love deformations. The derived surface strain measures are the classical ones: Naghdi’s surface strain measure generally and its well-known particular case for the Kirchhoff-Love deformations. With the bending strain measures comes a surprise: they are different from the traditional ones. For shear deformations our analysis provides a new tensor [Formula: see text], which is different from the widely used Naghdi’s bending strain tensor [Formula: see text]. In the particular case of Kirchhoff–Love deformations, the tensor [Formula: see text] reduces to a tensor [Formula: see text] introduced earlier by Anicic and Léger (Formulation bidimensionnelle exacte du modéle de coque 3D de Kirchhoff–Love. C R Acad Sci Paris I 1999; 329: 741–746). Again, [Formula: see text] is different from Koiter’s bending strain tensor [Formula: see text] (frequently used in this context). AMS 2010 classification: 74B99

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Uniqueness of Curvature Measures in Pseudo-Riemannian Geometry

Journal of Geometric Analysis ◽

10.1007/s12220-021-00702-4 ◽

2021 ◽

Author(s):

Andreas Bernig ◽

Dmitry Faifman ◽

Gil Solanes

Keyword(s):

Riemannian Manifolds ◽

Riemannian Geometry ◽

Riemannian Space ◽

Space Forms ◽

Type Formula ◽

Isometric Embeddings ◽

Curvature Measures ◽

Riemannian Space Forms

AbstractThe recently introduced Lipschitz–Killing curvature measures on pseudo-Riemannian manifolds satisfy a Weyl principle, i.e. are invariant under isometric embeddings. We show that they are uniquely characterized by this property. We apply this characterization to prove a Künneth-type formula for Lipschitz–Killing curvature measures, and to classify the invariant generalized valuations and curvature measures on all isotropic pseudo-Riemannian space forms.

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