Solution of quasi-periodic fracture problems by the representative cell method

Abstract

A general scheme for the solution of linear elastic quasi-periodic fracture problems is presented. The simplest type of such problems is characterized by a nonperiodic stress state in a domain with translational symmetry. Employing the discrete Fourier transform reduces the initial problem to a problem of a representative cell with speci®c boundary conditions which may be solved analytically or numerically. The procedure for solving the problem by the ®nite element method is developed. The suggested technique is employed for the solution of the problem of antiplane deformation of a strip weakened by a periodic array of arbitrary loaded cracks.

The analysis of localized effects in composites with periodic microstructure

“…In accordance with the RCM [1], the periodic composite shown in figure 1 region is divided into identical cells extending in three directions, labelled by (K 1 , K 2 , K 3 ), where…”

“…In general, these analyses are based on the combination of three approaches. In the first one, the representative cell method (RCM) [1,2] is used. Here, the composite is discretized into numerous identical cells, and the discrete Fourier transform is applied on the governing equations, interfacial and localization conditions.…”

Design of crack-resistant two-dimensional periodic cellular materials

“…We will be using a slightly modified version of stress localization developed in [Lipperman et al 2008a]. The unit loads solutions are obtained by the representative cell method [Nuller and Ryvkin 1980;Ryvkin and Nuller 1997], which enables to solve infinite and repetitive structures subjected to arbitrary loads, as opposed to periodic loads (see, for instance, [Ryvkin et al 1999;Fuchs and Ryvkin 2002;Fuchs et al 2004]). A discrete Fourier transform replaces the problem formulated for an infinite structure by an equivalent problem defined over the repetitive cell, albeit in transformed variables.…”

The Induced Stress Field in Cracked Composites by Heat Flow

“…As a result of the crack existence, the periodicity of the composite is lost, and consequently, a repeating unit cell (a representative volume element) does not exist. In the following, we present a method of solution that is based on the representative cell method [16], which is capable of establishing the full temperature and thermoelastic fields. To this end, let us consider a rectangular domain −D ≤ X 1 ≤ D, −H ≤ X 2 ≤ H of the composite, which includes the crack region.…”

“…This is carried out by utilizing the high-fidelity generalized method of cells (HFGMC), which has been fully described in [15]. In the framework of the macromechanical analysis, the representative cell method [16] is employed. According to this method, the composite domain is divided into several rectangular cells with respect to which the governing and constitutive equations are formulated.…”