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

Abstract

A rectangular cell of known boundary conditions is cut out from a medium containing the doubly periodic array of inclusions. The stress and strain relationship of the rectangular cell is obtained by using the classical boundary element methods. By matching the boundary condition requirements, the effective elastic properties of the doubly periodic array of inclusions can then be calculated. Numerical examples from the sub-domain boundary element method and the single domain boundary element method are compared and discussed. However, the present method cannot be readily extended to domains having circular or curved boundary parts.

Effective elastic properties of solids with irregularly shaped inclusions

“…9a, b with solid lines. Dong's numerical results (Dong 2006) are also Effective elastic properties by the ad hoc FEM analysis 237 marked in Fig. 9a, b with open circles.…”

“…However, sufficiently large samples behave homogeneously. Therefore, similar to doubly periodic inclusion problems (Dong 2006), a composite material containing disordered array of inclusions of irregular shapes discussed in this paper can also be considered as a homogeneous orthotropic material. Its constitutive relation is given as follows (Lekhnitskii 1963):…”

“…The elastic modulus and the Poisson ratio of the inclusion are taken as E I and t I ¼ t M ¼ 0:3, respectively. This problem has been analyzed by using the boundary element method (Dong 2006). In the present analysis, because of the symmetry of the geometry and the loading conditions, it is sufficient analyze only a quarter of the cell.…”

“…8b, c, respectively. At four different element number levels, comparisons of the effective elastic properties obtained by using the novel ad hoc hybrid-stress and standard four-node hybrid-stress finite element methods with Dong's numerical results (Dong 2006) are given in Table 1. It can be seen from Table 1 that the numerical solutions from the novel ad hoc hybrid-stress and standard four-node hybridstress finite element methods are in excellent agreement with those of Dong (2006); either from the number of elements or from the accuracy of results the novel ad hoc hybrid-stress finite element method is better than the standard four-node hybrid-stress finite element method, which confirms the observations in last paragraph of Sect.…”

“…Compared with the conventional FEM, the boundary integral equation method (BIEM) is a more efficient and accurate one for solving inclusion problems. The BIEM has been successfully applied for the solution of inclusion problems of various shapes (Chen 1994;Noda et al 2000Noda et al , 2003Dong 2006). However, the method also needs the fundamental solution, which is not easy-to-obtain or impossible-to-obtain for composite materials; and the boundary integral equation must be formulated for each inclusion, which might be inconvenient for solving problems containing many inclusions of irregular shapes.…”