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Research Papers

# Homogenization of Rough Two-Dimensional Interfaces Separating Two Anisotropic Solids

[+] Author and Article Information
Pham Chi Vinh1

Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science, 334, Nguyen Trai Street, Thanh Xuan, Hanoi 10000, Vietnampcvinh@vnu.edu.vn

Do Xuan Tung

Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science, 334, Nguyen Trai Street, Thanh Xuan, Hanoi 10000, Vietnam

1

Corresponding author.

J. Appl. Mech 78(4), 041014 (Apr 14, 2011) (7 pages) doi:10.1115/1.4003722 History: Received August 04, 2010; Revised January 22, 2011; Posted February 28, 2011; Published April 14, 2011; Online April 14, 2011

## Abstract

In this paper we have derived homogenized equations in explicit form of the linear elasticity theory in a two-dimensional domain with an interface highly oscillating between two straight lines, by using the homogenization method. First, the homogenized equation in the matrix form for generally anisotropic materials is obtained. Then, it is written down in the component form for specific cases when the materials are orthotropic, monoclinic with the symmetry plane at $X1=0$ and $X2=0$. Since these equations are in explicit form, they are significant in practical applications.

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## Figures

Figure 1

Two-dimensional domains Ω+ and Ω− have a very rough interface L expressed by equation X3=Ah(X1/λ)=Ah(y), where h(y) is a periodic function with period 1, and its maximum and minimum values are 0 and −1, respectively. The curve L oscillates between the straight lines X3=0 and X3=−A.

Figure 2

The interface L in the x1x3-plane

Figure 3

The tooth-comb interface L. L1:X1=a(−A<X3<0), L2:X1=a+b(−A<X3<0), L3:X3=0(0≤X1≤a), and L4:X3=−A(a≤X1≤a+b),a,b=const>0

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