We show convergence of a cell-centered finite volume discretization for linear elasticity. The discretization, termed the MPSA method, was recently proposed in the context of geological applications, ...where cell-centered variables are often preferred. Our analysis utilizes a hybrid variational formulation, which has previously been used to analyze finite volume discretizations for the scalar diffusion equation. The current analysis deviates significantly from the previous in three respects. First, additional stabilization leads to a more complex saddle-point problem. Second, a discrete Korn's inequality has to be established for the global discretization. Finally, robustness with respect to the Poisson ratio is analyzed. The stability and convergence results presented herein provide the first rigorous justification of the applicability of cell-centered finite volume methods to problems in linear elasticity.
A minimum problem for a class of functionals defined in Orlicz-Sobolev spaces is studied. As application, the existence and uniqueness of the weak solutions to Dirichlet problems for the Hencky ...deformation operator are obtained.
A generalization due to Sorin Mardare of the fundamental theorem of surface theory for surfaces with little regularity asserts that, if, for any p>2, the components of a positive-definite 2×2 ...symmetric matrix field in the space Wloc1,p and the components of another 2×2 symmetric matrix field in the space Llocp satisfy together the Gauss and Codazzi-Mainardi equations in a simply-connected open subset of R2, then there exists a surface defined in the three-dimensional Euclidean space E3 by an immersion with components in the space Wloc2,p, whose first and second fundamental forms are precisely the given matrix fields; besides, this surface is uniquely determined up to isometries in E3.
We establish here that a surface defined in this fashion varies continuously as a function of its two fundamental forms for several Fréchet topologies, which include in particular the above spaces Wloc1,p for the first fundamental form and Llocp for the second fundamental form, for any p>2.
Une généralisation à des surfaces peu régulières du théorème fondamental de la théorie de surfaces, due a Sorin Mardare, affirme que, pour tout p>2, si les composantes dans Wloc1,p d'un champ de matrices symétriques et définies positives d'ordre 2 et les composantes dans Llocp d'un champ de matrices symétriques d'ordre 2 satisfont ensemble les équations de Gauss et de Codazzi-Mainardi dans un ouvert simplement connexe de R2, alors il existe une surface définie dans l'espace euclidien tri-dimensionnel E3 par une immersion avec des composantes dans l'espace Wloc2,p, dont les première et deuxième formes fondamentales sont précisément les deux champs de matrices donnés ; de plus, cette surface est définie de façon unique, aux isométries près de E3.
On établit ici qu'une surface ainsi définie varie continûment en fonction de ses formes fondamentales pour plusieurs topologies de Fréchet, qui incluent en particulier pour tout p>2 les espaces Wloc1,p pour la première forme fondamentale et Llocp pour la seconde.
We establish the first partial regularity results for (strongly) symmetric quasiconvex functionals of linear growth on BD, the space of functions of bounded deformation. By Rindler's foundational ...work 64, symmetric quasiconvexity is the pivotal notion as regards sequential weak*-lower semicontinuity and hence for the existence of minima of the relaxed functionals on BD. The overarching main difficulty here is the lack of Korn's Inequality in the L1-setting, hereby implying that the BD-case is genuinely different from the study of variational integrals on BV. Unlike for superlinear growth, symmetric quasiconvex functionals, where we establish partial regularity by direct reduction to the full gradient case by Korn-type inequalities, such a reduction does not work in the linear growth case and identifies the latter as the only situation requiring a treatment on its own.
Nous établissons les premiers résultats de régularité partielles pour des fonctionnelles quasi-convexes (fortement) symétriques ayant une croissance linéaire sur BD, l'espace des fonctions dont la déformation est bornée. De part les travaux précurseurs de Rindler64, la quasi-convexité symétrique est la notion centrale relativement à la semi-continuité inférieure faible-* et donc pour l'existence des minima de fonctionnelles relaxées sur BD. La difficulté générale ici est due à l'absence de l'inégalité de Korn dans le cadre fonctionel L1, qui de ce fait révèle une différence fondamentale entre le cas BD et celui relatif à l'étude des intégrales variationnelles sur BV. Contrairement aux fonctionnelles quasi-convexes symétriques à croissance superlinéaire pour lesquelles nous établissons la régularité partielle via réduction directe au cas du gradient complet grâce aux inégalités de type Korn, une telle réduction ne peut être implémentée dans le cas de la croissance linéaire et identifie ce dernier comme étant la seule situation qui nécéssite un traitement particulier.
We prove constructive estimates for elastic plates modeled by the Reissner–Mindlin theory and made by general anisotropic material. Namely, we obtain a generalized Korn inequality which allows to ...derive quantitative stability and global H2 regularity for the Neumann problem. Moreover, in case of isotropic material, we derive an interior three spheres inequality with optimal exponent from which the strong unique continuation property follows.
Abstract
Korn’s inequality has been at the heart of much exciting research since its first appearance in the beginning of the 20th century. Many are the applications of this inequality to the ...analysis and construction of discretizations of a large variety of problems in continuum mechanics. In this paper, we prove that the classical Korn inequality holds true in
anisotropic Sobolev spaces.
We also prove that an extension of Korn’s inequality, involving non-linear continuous maps, is valid in such spaces. Finally, we point out that another classical inequality, namely Poincaré’s inequality, also holds in anisotropic Sobolev spaces.
A nonlinear Korn inequality in Rn provides an upper bound of an appropriate distance between two smooth enough immersions defined over an open subset Ω of Rn in terms of the corresponding distance ...between their metric tensors.
Under the assumption that Ω only satisfies the uniform interior cone property, we first establish such a nonlinear Korn inequality for immersions in the space W2,p(Ω), p>n, hence with metric tensors in the space W1,p(Ω); our point of departure is a crucial comparison theorem between solutions in W1,p(Ω) of Pfaff systems, which is due to the second author.
Under the assumptions that Ω is simply-connected and has a Lipschitz-continuous boundary, we then show that such immersions in W2,p(Ω) can be considered as well-defined functions, up to an isometric equivalence relation R(Ω), of their metric tensors in W1,p(Ω) if the Riemann curvature tensors of these tensors vanish in D′(Ω). We also show that the mapping defined in this fashion from the space W1,p(Ω) into the quotient set W2,p(Ω)/R(Ω) is locally Lipschitz-continuous.
Under the only assumption that Ω is simply-connected, we finally show that one can define an analogous mapping, this time acting from the space Wloc1,p(Ω), p>n, into the quotient set Wloc2,p(Ω)/R(Ω), and that this mapping is continuous when these spaces are equipped with their natural Fréchet topologies.
Une inégalité de Korn non linéaire dans Rn fournit une majoration d'une distance appropriée entre deux immersions suffisament régulières définies sur un ouvert Ω de Rn en fonction de la distance correspondante entre leurs tenseurs métriques.
Sous l'hypothèse que l'ouvert Ω satisfait seulement la propriété du cône intérieur uniforme, on établit pour commencer une telle inégalité de Korn non linéaire pour des immersions dans l'espace W2,p(Ω), p>n, donc avec des tenseurs métriques dans l'espace W2,p(Ω) ; notre point de départ est un théorème crucial de comparaison entre solutions dans l'espace W1,p(Ω) de systèmes de Pfaff, qui est dû au second auteur.
Sous l'hypothèse que l'ouvert Ω est simplement connexe et de frontière lipschitzienne, on montre ensuite que de telles immersions dans W2,p(Ω) peuvent être considérées comme des fonctions bien définies, à une relation R(Ω) d'équivalence isométrique près, de leurs tenseurs métriques dans W1,p(Ω) si les tenseurs de courbure de Riemann de ces derniers s'annulent dans D′(Ω). On montre aussi que l'application aussi définie de l'espace W1,p(Ω) dans l'ensemble quotient W2,p(Ω)/R(Ω) est localement lipschitzienne.
Sous la seule hypothèse que l'ouvert Ω est simplement connexe, on montre enfin que l'on peut définir une application analogue, agissant cette fois de l'espace Wloc1,p(Ω), p>n, dans l'ensemble quotient Wloc2,p(Ω)/R(Ω), et que cette application est continue lorsque ces espaces sont munis de leur topologies naturelles de Fréchet.
A nonlinear Korn inequality estimates the distance between two immersions from an open subset of
R
n
into the Euclidean space
R
k
,
k
⩾
n
⩾
1
, in terms of the distance between specific tensor fields ...that determine the two immersions up to a rigid motion in
R
k
. We establish new inequalities of this type in two cases: when k = n, in which case the tensor fields are the square roots of the metric tensor fields induced by the two immersions, and when k = 3 and n = 2, in which case the tensor fields are defined in terms of the fundamental forms induced by the immersions. These inequalities have the property that their constants depend only on the open subset over which the immersions are defined and on three scalar parameters defining the regularity of the immersions, instead of constants depending on one of the immersions, considered as fixed, as up to now.
In this paper, we study the asymptotic behavior of an
ε
-periodic 3D stable structure made of beams of circular cross-section of radius
r
when the periodicity parameter
ε
and the ratio
r
/
ε
...simultaneously tend to 0. The analysis is performed within the frame of linear elasticity theory and it is based on the known decomposition of the beam displacements into a beam centerline displacement, a small rotation of the cross-sections and a warping (the deformation of the cross-sections). This decomposition allows to obtain Korn type inequalities. We introduce two unfolding operators, one for the homogenization of the set of beam centerlines and another for the dimension reduction of the beams. The limit homogenized problem is still a linear elastic, second order PDE.
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