Homogeneization of a plane elastic arch
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 \textit{Ph. G. Ciarlet} [J. Ind. Math. Soc., New Ser. 40, 131-143 (1976; Zbl 0436.73081)], it is known that (1) has one and only one solution. For each fixed value of \(\epsilon\), let us denote \(\vec u^{\epsilon}\) the unique solution of (1), with EA, EI given by (4), (5), respectively. The main result of the paper shows that, as \(\epsilon\to 0\), we have: \((6)\quad \vec u^{\epsilon}\to \vec u\) in \(H^ 1_ 0(K)\times H^ 2_ 0(K)\)- weakly, where \(\vec u\) (called the homogeneized solution) is the unique solution of the following (homogeneized) problem: \[ To\quad find\quad u=(u_ 1,u_ 2)\in H^ 1_ 0(K)\times H^ 2_ 0(K),\quad s.t., \] \[ \int_{K}[q\gamma(\vec u)v'\!_ 1-Q\rho(uA\!)(v_ 1/R)']ds=\int_{K}f_ 1v_ 1ds\quad \forall \quad v_ 1\in H^ 1_ 0(K) \] \[ \int_{K}[-q\gamma(\vec u)(v_ 2/R)-Q\rho(u)v''\!_ 2]ds=\int_{K}f_ 2v_ 2ds\quad \forall \quad v_ 2\in H^ 2_ 0(K), \] where q, Q are constant coefficients which do not depend on \(\epsilon\). They only depend upon \(e_ 1\), \(e_ 2\), representing a suitable average of the periodic functions \(e_ 1\), \(e_ 2\). For explicit formulas for q, Q we refer to section 3.2 of the paper. Finally, let us mention that in the last sections of this paper a complete study of the first corrector term associated with the homogeneized solution is presented, and an example concerning the effective computation of the homogeneized solution is studied.
scientific article; zbMATH DE number 3854585