Homogeneization of a plane elastic arch
type: Item
instance of: scholarly article
title: Homogeneization of a plane elastic arch
zbMATH Open document ID: 0537.73078
DOI: 10.1007/BF00041082
author: Adam Lutoborski
author: J. J. Telega
published in: Journal of Elasticity
publication date: 1984-01-01T00:00:00Z
review text: By means of the energy method introduced by \textit{L. Tartar} [''Cours Peccot'', College de France (1977)], the homogeneization of a plane elastic arch is studied for which the stiffness function EA and the bending stiffness function EI are periodic, with period equal to a real parameter \(\epsilon\) which tends to zero. Let \(K=(0,\ell)\subset {\mathbb{R}}\), \(\ell>0\), be an interval of the real axis. The variational formulation of the problem studied in this paper is: \[ (1)\quad To\quad find\quad \vec u=(u_ 1,u_ 2)\in H^ 1_ 0(K)xH^ 2_ 0(K),\quad s.t. \] \[ \int_{K}[EA\gamma(\vec u)v'\!_ 1-EI\rho(\vec u)(v_ 1/R)']ds=\int_{K}f_ 1v_ 1ds\quad \forall \quad v_ 1\in H^ 1_ 0(K) \] \[ \int_{K}[-EA\gamma(\vec u)(v_ 2/R)-EI\rho(u)v''\!_ 2]ds=\int_{K}f_ 2v_ 2ds\quad \forall \quad v_ 2\in H^ 2_ 0(K), \] where \(\vec u\) is the unknown displacement vector, and \((f_ 1,f_ 2)\) is the external force vector, which is given in \(L^ 2(K)^ 2\). In (1), E, A, I, which are strictly positive functions of s, represent the Young's modulus of the arch, the area of the cross-section of the arch, and the inertia moment of this cross- section, respectively. \(\gamma\) (\(\vec u)\) is the extension deformation function, given by \((2)\quad \gamma(\vec u)=u'\!_ 1-(u_ 2/R),\) where (1/R) is the curvature of the arch, and \(\rho\) (\(\vec u)\) represents the change of curvature, that is, \((3)\quad \rho(\vec u)=-((u_ 1/R)+u'\!_ 2)'.\) In (1), (2), (3), the symbol ' denotes derivative with respect to s. Let \(\epsilon\) be a positive real parameter. In this paper, problem (1) is studied assuming that the coefficient-functions EA, EI are given \(\epsilon\)-periodic functions having the following form: \((4)\quad EA(s)=e_ 1(s/\epsilon),\quad s\in K, (5)\quad EI(s)=e_ 2(s/\epsilon),\quad s\in K,\) where \(e_ 1\), \(e_ 2\) are regular and strictly positive functions, defined on \(Y=[0,1],\) and it is assumed that they have been Y-periodically extended to the whole line \({\mathbb{R}}.\) By \texti…
Mathematics Subject Classification ID: 74S30
Mathematics Subject Classification ID: 74K15
Mathematics Subject Classification ID: 65Z05
Mathematics Subject Classification ID: 74E05
zbMATH DE Number: 3854585
zbMATH Keywords: variational weak formulation
zbMATH Keywords: energy method
zbMATH Keywords: homogeneization
zbMATH Keywords: plane elastic arch
reviewed by: Carlos Conca
MaRDI profile type: Publication
cites work: Q4121498
cites work: Solution of Interface Problems by Homogenization. I
cites work: Q4189539
cites work: Q4128127
cites work: Elastic-Plastic Torsion of Heterogeneous Cylindrical Bars
cites work: A penalty duality method for the Budiansky-Sanders shell model
cites work: Q5549989
cites work: Homogenization and bifurcation of perforated plates
cites work: Non-homogeneous media and vibration theory
full work available at URL: https://doi.org/10.1007/bf00041082
OpenAlex ID: W2013869680
description: scientific article; zbMATH DE number 3854585