Two-dimensional elastic analysis of doubly periodic circular holes in infinite plane

Abstract

Two-dimensional elastic analysis of doubly periodic circular holes in an infinite plane is given in this paper. Two cases of loading, remote tension and remote shear, are considered. A rectangular cell is cut from the infinite plane. In both cases, the boundary value problem can be reduced to a complex mixed one. It is found that the eigenfunction expansion variational method is efficient to solve the problem. Based on the deformation response under certain loading, the notched medium could be modeled by an orthotropic medium without holes. Elastic properties for the equivalent orthotropic medium are investigated, and the stress concentration along the hole contour is studied. Finally, numerical examples and results are given.

Effective elastic properties of doubly periodic array of inclusions of various shapes by the boundary element method

“…Based on these results, the elastic properties of the equivalent orthotropic medium can be achieved. The novelty of this paper is to combine the boundary element method and the rectangular cell model cut from a medium (Chen and Lee, 2002;Dong and Lee, 2005a) for a solution to the doubly periodic inclusion problems. The author is of the view that it might be difficult for the proposed method to be applied in a medium having circular or curved boundary parts.…”

“…tension, the implementation of their method might be quite complicated. The eigenfunction expansion variational method (Chen, 1983) was adopted to investigate doubly periodic circular holes in infinite plane (Chen and Lee, 2002). Nevertheless, it is still rather difficult to solve doubly periodic inclusion problems as all the above methods only tackle the inclusions of simple geometric shapes.…”

An iterative FE–BE method and rectangular cell model for effective elastic properties of doubly periodic anisotropic inclusion composites

“…Similar to 2D circular hole problems [13], the model as shown in Fig. 1(b) can be decomposed into two sub-models as shown in Fig.…”

“…This method needs to solve a linear system of equations to obtain the coefficients of the expansion along suitable doubly periodic functions. The eigenfunction expansion variational method (EEVM) [12] has been utilized to study doubly periodic circular holes in infinite plane [13], but it is difficult to solve the corresponding circular or elliptical inclusions. The mentioned researches above are only limited to the special inhomogeneities, e.g.…”

Effective elastic properties of solids with irregularly shaped inclusions

“…Among the works on the analytical solution, the representative ones that should be mentioned are those of Nemat-Nasser and Hori (1993), Meguid and Zhong (1997), Deng and Meguid (1999), Zhong and Meguid (1999), Jiang et al (2004), Shtrikman (1962, 1963) and later publications, Budiansky (1965), Eshelby (1957) and later publications, Chen and Lee (2002) as well as Buryachenko (2007). Analytical solutions are either limited to very simple geometries such as elliptical inclusions or require high level of mathematical competence.…”