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

Kazmerchuk, Yulija A.; Mel'nyk, Taras A.

doi:10.1007/s11072-009-0058-4

Cited by 5 publications

(2 citation statements)

References 17 publications

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“…In 29 one can find careful asymptotic analysis of a problem in domain with multiscale oscillating boundary. A rich collection of new results on asymptotic analysis of boundary‐value problems in thick multi‐structures appeared in the following papers 21, 20, 30–42. New results in domains with oscillating boundaries can be seen in 28, 43–49.…”

Section: Introductionmentioning

confidence: 99%

On the rate of convergence of solutions in domain with periodic multilevel oscillating boundary

Chechkin

D’Apice

Maio

2010

Math. Meth. Appl. Sci.

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In this paper we deal with the homogenization problem for the Poisson equation in a singularly perturbed domain\ud with multilevel periodically oscillating boundary. This domain consists of the body, a large number of thin cylinders\ud joining to the body through the thin transmission zone with rapidly oscillating boundary. Inhomogeneous Fourier\ud boundary conditions with perturbed coefficients are set on the boundaries of the thin cylinders and on the boundary\ud of the transmission zone. We prove the homogenization theorems and derive the estimates for the convergence of the\ud solutions

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Section: Introductionmentioning

confidence: 99%

On the rate of convergence of solutions in domain with periodic multilevel oscillating boundary

Chechkin

D’Apice

Maio

2010

Math. Meth. Appl. Sci.

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“…Trusdell starts from Signorini's publication [80]. Of recent popularity is the Signorini problem with so-called Signorini boundary conditions, which is analyzed by mathematicians who represent many countries and schools of thought [20,33,36,37,39,42,43,45,48,77,78,90].The paper is structured as follows. Its goal is to unite recent results on waves in materials described by the Signorini elastic potential.…”

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confidence: 99%

Evolution of the theory of nonlinear waves in Murnaghan and Signorini materials

Rushchitsky

2009

Int Appl Mech

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The development of the classical Signorini nonlinear elastic model is continued. Recent year's published results on a recently developed area of the theory of nonlinear waves are outlined from a unified position and analyzed. The paper consists of six sections: explanation of the features of the class of waves under consideration; detailed description of the Signorini model; general formulation of wave-propagation problems based on the Signorini potential; analysis of results on plane and cylindrical waves obtained with the Signorini model; comparison of these results with those obtained with the Murnaghan model; discussion of the possible lines for the development of the research.Introduction. After many decades of neglect, the Signorini model is addressed here to clarify its capabilities in analyzing plane waves. Antonio Signorini (1888Signorini ( -1963, an outstanding Italian scientist, is by no means forgotten in modern mechanics. He is mentioned and quoted repeatedly in a number of classical books on continuum mechanics. For example, Chapter 9 of the book [87] discusses and elaborates upon Signorini's idea of iterative solution of nonlinear boundary-value problems in elasticity. Trusdell starts from Signorini's publication [80]. Of recent popularity is the Signorini problem with so-called Signorini boundary conditions, which is analyzed by mathematicians who represent many countries and schools of thought [20,33,36,37,39,42,43,45,48,77,78,90].The paper is structured as follows. Its goal is to unite recent results on waves in materials described by the Signorini elastic potential. A feature of the associated class of nonlinear problems is that they originally dealt with materials described by the Murnaghan potential. A general view of the achievements in solving wave problems with this model is presented in the reviews [17,18,60,[68][69][70] and, partially, in the recent monographs [9,19,30,50]. Therefore, results produced by the Murnaghan model are included here only for the sake of coherence and consistency. The present paper is mainly based on the author's publications [15,[25][26][27][28][29][30][31][61][62][63][64][65][66][67][68][69][70][71][72][73][74][75][76] and consists of seven sections. The structure of the paper is described, features of the class of nonlinear elastic waves in materials described by elastic potentials are listed, and the main concepts are interpreted in Introduction. Section 1 describes the Signorini potential and gives some new comments on the Signorini constant, the Kelvin and Poynting effects, and Volterra distortions. Section 2 provides a general problem formulation for waves in nonlinear elasticity based on the Signorini potential and the transformation from the Eulerian to Lagrangian description of deformation, resulting in nonlinear wave equations in Lagrange coordinates. Section 3 preliminarily compares the nonlinear wave equations in the Murnaghan and Signorini models, including the solutions for longitudinal and transverse plane waves obtained with these models. Section...

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Homogenization of the Signorini boundary-value problem in a thick junction and boundary integral equations for the homogenized problem

Mel'nyk¹,

Nakvasiuk²,

Wendland³

2010

Math. Meth. Appl. Sci.

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We consider a mixed boundary-value problem for the Poisson equation in a thick junction X e which is the union of a domain X 0 and a large number of e-periodically situated thin cylinders. The non-uniform Signorini conditions are given on the lateral surfaces of the cylinders. The asymptotic analysis of this problem is done as e → 0, i.e. when the number of the thin cylinders infinitely increases and their thickness tends to zero. We prove a convergence theorem and show that the non-uniform Signorini boundary conditions are transformed in the limiting variational inequalities in the region that is filled up by the thin cylinders as e → 0. The convergence of the energy integrals is proved as well. The existence and uniqueness of the solution to this non-standard limit problem is established. This solution can be constructed by using a penalty formulation and successive iteration. For some subclass, these problems can be reduced to an obstacle problem in X 0 and an appropriate postprocessing. The equations in X 0 finally are also treated with boundary integral equations.

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

Cited by 5 publications

References 17 publications

On the rate of convergence of solutions in domain with periodic multilevel oscillating boundary

On the rate of convergence of solutions in domain with periodic multilevel oscillating boundary

Evolution of the theory of nonlinear waves in Murnaghan and Signorini materials

Homogenization of the Signorini boundary-value problem in a thick junction and boundary integral equations for the homogenized problem

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