tailieunhanh - Effective boundary condition for the reflection of shear waves at the periodic rough boundary of an elastic body

We present a homogenization method to treat the problem of the reflection of waves at the free boundary of an elastic body, the edge being structured periodically at the subwavelength scale. The problem is considered for shear waves and the wave equation in the time domain is considered. | Vietnam Journal of Mechanics, VAST, Vol. 40, No. 4 (2018), pp. 303 – 323 DOI: EFFECTIVE BOUNDARY CONDITION FOR THE REFLECTION OF SHEAR WAVES AT THE PERIODIC ROUGH BOUNDARY OF AN ELASTIC BODY Agn`es Maurel1 , Jean-Jacques Marigo2,∗ , Kim Pham3 1 Institut Langevin, CNRS, ESPCI ParisTech, France 2 Laboratoire de M´ecanique du Solide, CNRS, Ecole Polytechnique, France 3 Unit´e de M´ecanique, ENSTA ParisTech, France ∗ E-mail: marigo@ Received Frebuary 07, 2018 Abstract. We present a homogenization method to treat the problem of the reflection of waves at the free boundary of an elastic body, the edge being structured periodically at the subwavelength scale. The problem is considered for shear waves and the wave equation in the time domain is considered. In the homogenized problem, a boundary condition at an equivalent flat edge is obtained, which links the normal stress to its derivatives, instead of the usual traction free condition. The problem of the position of the equivalent flat boundary with respect to the real roughnesses is addressed and this is done considering the equation of energy conservation in the homogenized problem and considering the accuracy of the homogenized solution when compared to the real one. Keywords: homogenization method, reflection of waves, rough free boundary, subwavelength scale, shear waves, energy conservation. 1. INTRODUCTION The problem of the propagation of waves in complex geometries most often cannot be solved analytically and it requires a numerical resolution. Among the different source of complexity, the existence of very different length scales is the worth in terms of numerical computations, at least when one expects that both the smallest and the largest scales have an impact on the wave behavior; this is because the mesh has to resolve the rapid variations associated to the small scale while the computational domain has be sized with the largest scale. If the