Homogenization of an interface highly oscillating between two concentric ellipses

The main purpose of this paper is to find the homogenized equation and the associate continuity conditions in the explicit form of a boundary-value problem in a twodimensional domain with an interface oscillating rapidly between two concentric ellipses. This boundary-value problem originates from various problems in practical applications. | Vietnam Journal of Mechanics, VAST, Vol. 34, No. 2 (2012), pp. 113 – 121 HOMOGENIZATION OF AN INTERFACE HIGHLY OSCILLATING BETWEEN TWO CONCENTRIC ELLIPSES Do Xuan Tung, Pham Chi Vinh, Nguyen Kim Tung Hanoi University of Science, VNU Abstract. The main purpose of this paper is to find the homogenized equation and the associate continuity conditions in the explicit form of a boundary-value problem in a twodimensional domain with an interface oscillating rapidly between two concentric ellipses. This boundary-value problem originates from various problems in practical applications. By the homogenization method and following the technique presented recently by Vinh and Tung [P. C. Vinh and D. X. Tung, Mech. Res. Comm. 37 (2010), 285-288; P. C. Vinh, D. X. Tung, ASME J. Appl. Mech., 78 (2011), 041014-1; P. C. Vinh and D. X. Tung, Acta Mech. 218 (2011), 333-348], the homogenized equation and the associate continuity conditions in the explicit form are derived. Key words: Interfaces oscillating highly between two concentric ellipses, homogenization method, homogenized equation. 1. INTRODUCTION Boundary-value problems in domains with rough boundaries or interfaces is closely related to various practical problems such as scattering of elastic waves at rough boundaries and interfaces [1], transmission and reflection of waves on rough interfaces [2, 3, 4], mechanical problems concerning the plates with densely spaced stiffeners [5], flows over rough walls [6] and so on. When the amplitude (height) of the roughness is much small comparison with its period, the problems are usually analyzed by perturbation methods. When the amplitude is much large than its period, . the boundaries and interfaces are very rough, the homogenization method [7, 8, 9] is required. In [10], Nevard and Keller investigated a boundary-value problem in a two-dimensional domain with an interface highly oscillating between two straight lines (see Fig. 1), namely (σUx )x + (σUz )z − λU = f (x, z), (x,

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