European Journal of Mechanics A/Solids 21 (2002) 465–481 A micromechanically based couple-stress model of an elastic orthotropic two-phase composite Frederic Bouyge a , Iwona Jasiuk a , Stéphane Boccara a , Martin Ostoja-Starzewski b,∗ a The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0405, USA b Department of Mechanical Engineering, McGill University, Montréal, QC H3A 2K6, Canada Received 9 November 2000; revised and accepted 18 October 2001 Abstract We determine couple-stress moduli and characteristic lengths of a two-dimensional matrix-inclusion composite, with inclusions arranged in a periodic square array and both constituents linear elastic of Cauchy type. In the analysis we replace this composite by a homogeneous planar, orthotropic, couple-stress continuum. A generalization of the original Mindlin’s (1963) derivation of field equations for such a continuum results in two (not just one!) characteristic lengths. We evaluate the couple- stress properties from the response of a unit cell under several types of boundary conditions: displacement, displacement- periodic, periodic and mixed, and traction controlled. In the parametric study we vary the stiffness ratio of both phases to cover a range of different media from nearly porous materials through composites with very stiff inclusions. We find that the aforementioned boundary conditions result in hierarchies of orthotropic couple-stress moduli, whereas both characteristic lengths are fairly insensitive to boundary conditions, and fall between 0.12% and 0.22% of the unit cell size for the inclusions’ volume fraction of 18%.  2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved. 1. Introduction Cosserat-type (also called micropolar or microcontinuum) theories (Cosserat and Cosserat, 1909) include microstructural length scale in their formulation, which is absent in classical continuum theories. These theories have been proposed for situations when the materials microstructure has a comparable scale to the materials’ dimensions thus giving rise to sharp gradients of dependent fields. While capable of grasping microstructural effects better, the main challenge of this approach is the determination of Cosserat constitutive coefficients. Recent few years have witnessed increasing activity in that direction. Forest and Sab (1998) and Forest (1998) proposed a methodology for derivation of an effective, homogeneous Cosserat-type continuum for a periodic heterogeneous Cauchy- type continuum. Their approach involves the extension of the classical homogenization method (e.g., Sanchez-Palencia and Zaoui (1987)). Using a finite element method they also amply demonstrate the advantage of replacing the actual Cauchy- type microstructure by the Cosserat-type continuum in that such a model requires a much smaller number of degrees of freedom. Other related works include (Pideri and Seppecher, 1997; de Buhan and Surdet, 2000; Luciano and Willis, 2001; Peerlings and Fleck, 2001). The couple-stress model, which is the simplest and most restrictive case of all the Cosserat models, was recently studied by us in (Ostoja-Starzewski et al., 1999) and (Bouyge et al., 2001). We investigated a planar, periodic, effectively isotropic, two-phase composite (with a triangular arrangement of inclusions) with linear elastic constituents of classical Cauchy-type. In * Correspondence and reprints. E-mail address: martin.ostoja@mcgill.ca (M. Ostoja-Starzewski). 0997-7538/02/$ – see front matter  2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved. PII:S0997-7538(01)01192-5