Boundary Homogenization and Reduction of Dimension in a Kirchhoff–Love Plate

Blanchard, Dominique; Gaudiello, Antonio; Mel'nyk, Taras A.

doi:10.1137/070685919

Cited by 49 publications

(26 citation statements)

References 10 publications

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“…Clearly, one could push forward presented work by considering some other important questions such as small-amplitude homogenization and G-closure problem for stationary plate equation, problem of boundary homogenization in more general terms [6], or considering homogenization theory for vibrating plates (see [4]). We leave this open for some future work.…”

Section: Discussionmentioning

confidence: 99%

Homogenization of Elastic Plate Equation∗

Burazin

Jankov

Vrdoljak

2018

Mathematical Modelling and Analysis

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Abstract. We are interested in general homogenization theory for fourth-order elliptic equation describing the Kirchhoff model for pure bending of a thin solid symmetric plate under a transverse load. Such theory is well-developed for second-order elliptic problems, while some results for general elliptic equations were established by Zhikov, Kozlov, Oleinik and Ngoan (1979). We push forward an approach of Balenović (1999, 2000) by proving a number of properties of H-convergence for stationary plate equation.

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Section: Discussionmentioning

confidence: 99%

Homogenization of Elastic Plate Equation∗

Burazin

Jankov

Vrdoljak

2018

Mathematical Modelling and Analysis

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“…The only result we can mention on that score relates to a model problem, namely, for the Poisson equation with inequality type boundary condition at the crack faces [10]. We choose U = (θ, 0), where θ is a smooth function equal to 1 near (1, 0) with the support in a neighborhood of the point (1,0). Also assume that f = 0, g = 0 in a vicinity of the point (1, 0).…”

Section: Invariant Integralsmentioning

confidence: 99%

“…In fact, we should consider multi-domains with inequality type glue conditions (see [11,12]). As for models describing junctions of elastic bodies with equality type glue conditions we refer to [1] and [16] with the references quoted therein; elastic plate models can be found in [3]. Moreover, it turned out that many other practical problems should be described by crack models with overlapping domains.…”

Section: Introductionmentioning

confidence: 99%

Crack on the boundary of two overlapping domains

Gaudiello

Khludnev

2009

Z. Angew. Math. Phys.

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In this paper, we consider an overlapping domain problem for two elastic bodies. A glue condition of an equality-type is imposed at a given line. Simultaneously, a part of this line is considered to be a crack face with an inequality-type boundary condition describing mutual non-penetration between crack faces. Variational and differential formulations of the problem are considered. We prove a differentiability of the energy functional in the case of rectilinear cracks and find a formula for invariant integrals. Passage to the limit is justified provided that the rigidity of the body goes to infinity. (2000). 49J40 · 49K10 · 74R10. Mathematics Subject ClassificationKeywords. Crack with non-penetration · Derivative of energy functional · Overlapping domain. IntroductionIn solid mechanics, classical crack problems are characterized by linear boundary conditions imposed at the crack faces. It is well known that such linear models allow the opposite crack faces to penetrate each other which leads to inconsistency with practical situations. Since the beginning of 1990, the crack theory with non-penetration conditions has been under active study. This theory is characterized by inequality type boundary conditions at the crack faces. The book [6] contains results for crack models with nonpenetrations for different constitutive laws. In particular, 2D and 3D models as well as plate and shell models with inequality type boundary conditions are analyzed. After publication of this book, new approaches and trends in the study of non-linear crack models with non-penetration have been developed. For example, a problem of differentiation of the energy functional with respect to crack perturbations is solved in a general setting, smooth and fictitious domain methods are proposed, invariant integrals are constructed in cases of different geometrical situations, etc. We refer the reader to publications [5,[7][8][9]15,[17][18][19]. The latest results related to the crack growth for linear crack models (i.e. with linear boundary conditions at the crack faces) can be found in [2,4].There are other problems formulated in non-smooth domains with inequality type boundary conditions. It is known that contact problems for bodies of different dimensions are described by models similar to those of crack models with non-penetration. For these contact problems, the equilibrium equations are satisfied in cracked domains, and the boundary conditions are of inequality type. In fact, we should consider multi-domains with inequality type glue conditions (see [11,12]). As for models describing junctions of elastic bodies with equality type glue conditions we refer to [1] and [16] with the references quoted therein; elastic plate models can be found in [3]. Moreover, it turned out that many other practical problems should be described by crack models with overlapping domains. This overlapping approach is applicable for description of the subduction phenomenon of tectonic plates as well as for the slipping phenomenon of ice plates, and also for the ...

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“…In recent years, a rich collection of new results on asymptotic analysis of boundary-value problems in thick multi-structures is appeared (see [1]- [8]). …”

Section: Introduction and Statement Of The Problemmentioning

confidence: 99%

Homogenization of the signorini boundary-value problem in a thick plane junction

Kazmerchuk

Mel'nyk

2009

Nonlinear Oscill

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We consider a mixed boundary-value problem for the Poisson equation in a plane thick junction Ω ε which is the union of a domain Ω 0 and a large number of ε−periodically situated thin rods. The nonuniform Signorini conditions are given on the vertical sides of the thin rods. The asymptotic analysis of this problem is made as ε → 0, i.e., when the number of the thin rods infinitely increases and their thickness tends to zero. With the help of the integral identity method we prove the convergence theorem and show that the nonuniform Signorini conditions are transformed (as ε → 0) in the limiting variational inequalities in the region that is filled up by the thin rods in the limit passage. The existence and uniqueness of the solution to this non-standard limit problem is established. The convergence of the energy integrals is proved as well.

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Boundary Homogenization and Reduction of Dimension in a Kirchhoff–Love Plate

Cited by 49 publications

References 10 publications

Homogenization of Elastic Plate Equation∗

Homogenization of Elastic Plate Equation∗

Crack on the boundary of two overlapping domains

Homogenization of the signorini boundary-value problem in a thick plane junction

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