Asymptotic analysis for a stiff variational problem arising in mechanics.

*(English)*Zbl 0723.73011The paper deals with the homogenization of periodic heterogeneous media. More precisely, it studies, from the mathematical point of view, the small vibrations of a solid-fluid mixture, the geometric structure of which is \(\epsilon\) Y-periodic \((\epsilon \ll 1)\). In the homogenization process, the solution is searched, when \(\epsilon\) tends to zero, in the form of a double scale asymptotic expansion in functions of \(\vec x\) (the standard macroscopic variable) and \(\vec y=\vec x/\epsilon\) (the microscopic ones), Y-periodic with respect to \(\vec y\). The case of a periodic mixture of an elastic solid and a compresssible viscous fluid is considered in the complex situation in which the elasticity coefficients are O(1), with respect to \(\epsilon\), whereas the viscosity coefficients are small as \(\epsilon^ 2.\)

The formal analysis and the determination of the bulk behaviour have been performed some years ago [Th. Levy, Rech. Math. Appl. 4, 206-222 (1987; Zbl 0654.76090)]. It emphasizes very different results according to whether or not the total fluid region is connected. But, the validation of the formal analysis, that means the proof of the convergence of the problem in \(\epsilon\) to the homogenized problem, remained an open question.

The main result of this paper is the demonstration of the convergence in the above situation. The formal limit of the displacement in the mixture does not depend on the local variable \(\vec y\) if the fluid parts of the mixture are not connected, whereas it depends on the local variable if the fluid parts are connected. Thus the problem is a unusual homogenization problem referred to as singular homogenization problem. Furthermore, the energy method, that can be used elsewhere, is not flexible enough to handle the present situation. In the both cases of connectedness of the fluid phase, a convergence theorem is proved.

The formal analysis and the determination of the bulk behaviour have been performed some years ago [Th. Levy, Rech. Math. Appl. 4, 206-222 (1987; Zbl 0654.76090)]. It emphasizes very different results according to whether or not the total fluid region is connected. But, the validation of the formal analysis, that means the proof of the convergence of the problem in \(\epsilon\) to the homogenized problem, remained an open question.

The main result of this paper is the demonstration of the convergence in the above situation. The formal limit of the displacement in the mixture does not depend on the local variable \(\vec y\) if the fluid parts of the mixture are not connected, whereas it depends on the local variable if the fluid parts are connected. Thus the problem is a unusual homogenization problem referred to as singular homogenization problem. Furthermore, the energy method, that can be used elsewhere, is not flexible enough to handle the present situation. In the both cases of connectedness of the fluid phase, a convergence theorem is proved.

Reviewer: Th.Lévy (Paris)

##### MSC:

74E05 | Inhomogeneity in solid mechanics |

74S30 | Other numerical methods in solid mechanics (MSC2010) |

74P10 | Optimization of other properties in solid mechanics |

35B40 | Asymptotic behavior of solutions to PDEs |

35B27 | Homogenization in context of PDEs; PDEs in media with periodic structure |

35B25 | Singular perturbations in context of PDEs |

74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |

76M30 | Variational methods applied to problems in fluid mechanics |