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Asymptotic analysis of Poisson’s equation in a thin domain and its application to thin-walled elastic beams and tubes. (English) Zbl 0935.35038

Let \(\omega^t\subset \mathbb R^2\) be a “profile” – a domain (open and connected set) with one of its characteristic dimensions (thickness) of order \(t>0\), which is assumed to be very small in comparison with the other one. \(\gamma^t\) is the boundary of \(\omega^t\), which is smooth enough \(\nu^t=(\nu^t_1,\nu^t_2)\) is the outward unit vector and \(x^t=(x^t_1,x^t_2)\) is a generic point of \(\overline{\omega}^t\). The following problems, associated to the Poisson’s equation in \(\omega^t\) with Dirichlet or Neumann boundary conditions, are considered: \[ -\Delta^t\psi^t=F^t\;\text{in}\;\omega^t,\quad \psi^t=0\;\text{on} \gamma^t,\quad -\Delta^t\phi^t=\text{div}^tf^t\;\text{in}\;\omega^t,\quad \partial^t_\nu\phi^t=-f^t\cdot \nu^t\;\text{on} \gamma^t, \]
\[ \int_{\omega^t}\phi^t\,dx^t=0,\quad -\Delta^t\eta^t=G^t\;\text{in} \omega^t,\quad \partial^t_\nu\eta^t=0\;\text{on}\;\gamma^t,\quad \int_{\omega^t}\eta^t\,dx^t=0, \] where \(\partial^t_\alpha,\partial^t_{\alpha\beta},\partial^t_\nu,\dots\) are differential operators \(\partial/\partial x^t_\alpha\), \(\partial^2/\partial x^t_\alpha\partial x^t_\beta\), \(\partial/\partial\nu^t,\dots;\) \[ \begin{aligned} & \Delta^t\psi^t=\partial^t_{11}\psi^t+\partial^t_{22}\psi^t,\nabla^t\phi^t=(\partial^t_1\phi^t,\partial^t_2\phi^t),\quad F^t\in L^2(\omega^t),\\ & f^t\in H(\text{div}^t,\omega^t),H(\text{div}^t,\omega^t)=\{f^t\in[L^2(\omega^t)]^2:|\text{div}^tf^t|\in L^2(\omega^t)\},\\ & |f^t|_{\text{div}^t,\omega^t}=[|f^t|^2_{0,\omega^t}+|\text{div}^tf^t|^2_{0,\omega^t}]^2,\quad G^t\in L^2(\omega^t),\;\int_{\omega^t}G^t\,dx^t=0.\end{aligned} \]
The main goal of this paper is to study the behavior of the solution \(\psi^t\), \(\varphi^t\) and \(\eta^t\) of the above problems as \(t\) becomes very small. In the first part of the paper it is considered a particular class of open and closed single-hollowed profiles \(\omega^t\) having a smooth centerline defined by a plane curve of class \(C^2\) and whose boundary is smooth enough – only four corners are allowed in the open profile and no corners in the closed profile [P. G. Ciarlet, H. Le Dret and R. Nzengwa, J. Math. Pures Appl. (9) 68, No. 3, 261–295 (1989; Zbl 0661.73013); (2) H. Le Dret, Problèmes variationnels dans les multi-domaines. Modélisation des jonctions et applications. Recherches en Mathématiques Appliquées. 19. Paris etc.: Masson (1991; Zbl 0744.73027)]. It is made a change of variable to a reference domain that transforms above problems on other, which have the thickness \(t\) as an explicit small parameter [J. L. Lions, Perturbations singulières dans les problèmes aux limites et en contrôle optimal. Lect. Notes Math. 323. Berlin etc.: Springer (1973; Zbl 0268.49001)].
In the second part of the paper the limit values of torsion, warping and Timoshenko’s functions and constants are obtained that could be used in the open and closed thin-elastic beam theories [J. T. Oden and E. A. Ripperger, Mechanics of elastic structures. 2nd ed. New York: McGraw-Hill (1981) and B. Z. Vlassov, Pièces longues en voiles minces. 2nd ed. Paris: Eyrolles (1962)].

MSC:

35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35B40 Asymptotic behavior of solutions to PDEs
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