Homogenization of very rough two-dimensional interfaces separating two dissimilar poroelastic solids with time-harmonic motions

Abstract

The homogenization of a very rough three-dimensional interface separating two dissimilar isotropic poroelastic solids with time-harmonic motions was considered by Gilbert and Ou (Acoustic wave propagation in a composite of two different poroelastic materials with a very rough periodic interface: A homogenization approach. Int J Multiscale Comput Eng 2003; 1(4): 431–440). The homogenized equations have been derived; however, they are still in implicit form. In this paper, the homogenization of a very rough two-dimensional interface separating two dissimilar generally anisotropic poroelastic solids with time-harmonic motions is investigated. The main aim of the investigation is to derive homogenized equations in explicit form. By employing the homogenization method, along with the matrix formulation of the poroelasticity theory, the explicit homogenized equations have been derived. Since these equations are totally explicit, they are very useful in solving practical problems. As an example proving this, the reflection and transmission of SH waves at a very rough interface of the tooth-comb type is considered. The closed-form analytical expressions of the reflection and transmission coefficients are obtained. Based on these expressions, the dependence of the reflection and transmission coefficients on some parameters is examined numerically.

Homogenization of very rough three-dimensional interfaces for the poroelasticity theory with Biot's model

“…When the motion of the poroelastic solids is the same along the direction perpendicular to the plane of right section of the very rough cylindrical interface, the problem is reduced to the homogenization of a two-dimensional very rough interface which is the right section (directrix) of the very rough cylindrical interface. Therefore, this paper can be considered as an extension of the investigation by Vinh et al [10].…”

“…(17) (corresponding to Biot's model) are not equal to the matrices A ik , B, D and E, respectively, in Eq. (27) in Vinh et al [10] (corresponding to Auriault's model), in general.…”

“…V does not depend on x 2 , the homogenized equation 17is simplified to Eq. (27) in Vinh et al [10]. It should be noted that the matrices A ik , B, D and E in Eq.…”

“…Using the matrix formulation (not the component formulation) of theories, Vinh and his coworkers carried out the homogenization of two-dimensional very rough interfaces and the explicit homogenized equations have been obtained for the elasticity theory [4][5][6][7], for the piezoelectricity theory [8], for the micropolar elasticity [9] and for the poroelasticity with Auriault's model for time-harmonic motions [10].…”

Reflection and transmission of P-waves at a very rough interface between two isotropic elastic solids

“…Most of the studies considered slightly rough boundaries and interfaces, see for examples, works [1,2], and the perturbation method [3] was employed to analyze the problems. However, due to the mathematical complexity caused by strong roughness of boundaries and interfaces, there are few studies concerning the reflection and transmission of waves at very rough boundaries and interfaces [4,5]. The traditional formulation of this problem leads to boundary integral equations whose numerical solution is unstable due to rapid oscillation of rough boundaries and interfaces.…”