## Asymptotic modeling of the elastodynamics of a multi-structure.(English)Zbl 0777.73033

Summary: We consider a structure consisting of two parts: a three-dimensional linearly elastic body and a linearly elastic plate. The plate is inserted into the three-dimensional body. We perform an asymptotic analysis of the time-dependent behavior of the structure during the time-interval $$[0,T]$$ when the thickness of the plate goes to zero. Under specific assumptions on the data (Lamé constants, mass densities, loads), we establish the existence of a limit problem. This limit problem is a system of coupled partial differential equations posed over a three-dimensional body with a slit and the middle surface of the plate. Strong convergence in $$L^ 2(0,T;H^ 1)$$ of the time-dependent displacements (with appropriate scaling) is proved.

### MSC:

 74H45 Vibrations in dynamical problems in solid mechanics 74K20 Plates 74B05 Classical linear elasticity