Hindawi Publishing Corporation Advances in Mechanical Engineering Volume 2013, Article ID 942073, 10 pages http://dx.doi.org/10.1155/2013/942073 Research Article Mixed Volume and Boundary Integral Equation Method with Elliptical Inclusions and a Circular/Elliptical Void Jung-Ki Lee, Heung-Soap Choi, and Young-Bae Han Department of Mechanical and Design Engineering, Hongik University, 2639 Sejong-Ro, Jochiwon-Eup, Sejong City 339-701, Republic of Korea Correspondence should be addressed to Jung-Ki Lee; inq3jkl@wow.hongik.ac.kr Received 31 May 2013; Accepted 7 September 2013 Academic Editor: Indra Vir Singh Copyright © 2013 Jung-Ki Lee et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A mixed volume and boundary integral equation method (mixed VIEM-BIEM) is used to calculate the plane elastostatic feld in an unbounded isotropic elastic medium containing multiple isotropic/orthotropic elliptical inclusions of arbitrary orientation and a circular/elliptical void subjected to remote loading. In order to investigate the infuence of a circular/elliptical void on the interfacial stress feld, a detailed analysis of the stress feld at the interface between the matrix and the central isotropic/orthotropic inclusion is carried out for the square packing of eight inclusions and one void, taking into account diferent values for the orientation angles and concentration of the inclusions. Te mixed method is shown to be very accurate and efective for investigating the local stresses in composites containing isotropic/anisotropic fbers and a circular/elliptical void. 1. Introduction A micrograph of the cross section of a phosphate glass fber/ polymer composite is shown in Figure 1 [1]. Figure 1 indicates that the fbers are close to ellipses, and the major axis of the elliptical fbers is not aligned in any one direction. If voids are caused by manufacturing and/or service induced defects, they may have a signifcant efect on the local stresses in the composite. A number of analytical techniques are available for solv- ing stress analysis of inclusion problems when the geometry of the inclusions is simple (i.e., cylindrical, spherical, or ellipsoidal) and when they are well separated [2–4]. However, these approaches cannot be applied to more general problems where the inclusions are of arbitrary shape and their con- centration is high. Tus, the stress analysis of heterogeneous solids ofen requires the use of numerical techniques based on the fnite element method (FEM) or boundary element method (BIEM). Unfortunately, both methods encounter limitations in dealing with problems involving infnite media or multiple inclusions. However, it has been demonstrated that a recently developed numerical method based on a volume integral formulation can overcome such difculties in solving a large class of inclusion problems [5–10]. One advantage of the volume integral equation method (VIEM) over the boundary integral equation method (BIEM) is that it does not require the use of Green’s functions for both the matrix and the inclusions [11, 12]. In addition, the VIEM is not sensitive to the geometry or concentration of the inclusions. Moreover, in contrast to the fnite element method (FEM), where the full domain needs to be discretized, the VIEM requires discretization of the inclusions only [13, 14]. However, the VIEM cannot be directly applied to prob- lems involving voids since the feld quantities are undefned in the domain of the integral equation. By contrast, problems involving voids can be most efectively solved using the BIEM. However, if the material contains a combination of elliptical inclusions and a circular/elliptical void, it is most advantageous to apply a combination of the two methods [7– 9]. Terefore, in this paper, the mixed volume and boundary integral equation method (mixed VIEM-BIEM) developed by Lee et al. [7–9] is used to calculate the elastic feld in com- posites consisting of an isotropic matrix containing multiple isotropic/anisotropic elliptical inclusions and a circular/ellip- tical void subjected to remote uniaxial tension. Te elliptical inclusions are assumed to have diferent orientations relative to the loading direction. In order to investigate the infuence