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Reinforced and honey-comb strucures. (English) Zbl 0656.35031
The aim of this paper is to describe the behaviour of large very thin space structures. In such structures the material is periodically distributed and the period \(\epsilon\) is small compared with the global dimensions. Moreover the thickness of the material \(\epsilon\) \(\delta\) is much smaller than the period i.e. \(\delta\) is another small parameter. In a period of reference the material can be distributed along layers - honeycomb structues - or along beams - reinforced structures. The system considered is a stationary heat equation with zero Neumann boundary conditions on the boundary of the holes.
In a first step, \(\delta\) is fixed and \(\epsilon\) \(\to 0\). The periodic distribution is dealt with by a classic homogenization method in domains with holes. The homogenized coefficients are obtained by solving a partial differential system on a period of reference \(Y=[0,1]^ 3.\)
The next step is the study of the homogenized solution in terms of the second small parameter \(\delta\). This is done by means of a priori estimates, decomposition of the period into the \(\delta\)-thick layers or beams which compose it and dilatation of these layers or beams in order to transform them into the cell Y. Then it is possible to pass to the limit in these dilatated elements of the structure and the final result is a mere superposition of the results obtained in the different directions. If a complex structure with many oblique bars is considered, it is possible to work on each direction independently of the others and add the different limits thus obtained. All calculations are made on the peroiod of reference.
The mathematical method we use enables us to describe the global behaviour of these structures without making long and expensive computation (due to the rapid oscillations of the coefficients and to the small thickness of the material). It is important to remark that the limit global coefficients are explicit expressions of the coefficients of the initial problem.
More recently the same authors extended this study to elasticity problems, to structures such towers or gridworks, to eigenvalue problems.

35J25 Boundary value problems for second-order elliptic equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35C20 Asymptotic expansions of solutions to PDEs
35B40 Asymptotic behavior of solutions to PDEs