Proc. R. Soc. A (2011) 467, 2896–2911 doi:10.1098/rspa.2010.0660 Published online 11 May 2011 Stress and couple-stress invariance in non-centrosymmetric micropolar planar elasticity BY HADY JOUMAA* AND MARTIN OSTOJA-STARZEWSKI Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA The stress-invariance problem for a chiral (non-centrosymmetric) micropolar material model is explored in two different planar problems: the in-plane and the anti-plane problems. This material model grasps direct coupling between the Cauchy-type and Cosserat-type (or micropolar) effects in Hooke’s law. An identical strategy of invariance is set for both problems, leading to a remarkable similarity in their results. For both problems, the planar components of stress and couple-stress undergo strong invariance, while their out-of-plane counterparts can only attain weak invariance, which restricts all compliance moduli transformations to a linear type. As an application, when heterogeneous (composite) materials are subjected to weak invariance, their effective (volume-averaged) compliance moduli undergo the same linear shift as that of the moduli of the local phases forming the material, independently of the microstructure, geometry and phase distribution. These analytical results constitute a valuable means to validate computational procedures that handle this particular type of material model. Keywords: stress invariance; micropolar elasticity; couple-stress; Cosserat model; chiral material; composite materials 1. Introduction The stress-invariance problem investigates the possibility of modifying the stiffness constants of a heterogeneous elastic body of arbitrary shape without altering the stress field within this deformed body, provided it is subjected to static traction boundary conditions throughout its entire boundary. The schematic layout of figure 1 illustrates the concept. This invariance was first recognized to hold in planar elasticity (Cherkaev et al. 1992; Thorpe & Jasiuk 1992), and subsequently explored and generalized to various situations (e.g. Dundurs & Markenscoff 1993; Moran & Gosz 1994; Norris 1999; Hu & Weng 2001; Jasiuk & Ostoja-Starzewski 2003). Reviews of this subject and related topics appeared in Milton (2002), Ostoja-Starzewski (2008) and Jasiuk (2009). One extension of the stress-invariance problem focused on the Cosserat (micropolar) elasticity (Ostoja-Starzewski & Jasiuk 1995). As is well known, theory grants a micropolar particle six degrees of freedom (three translations *Author for correspondence (hjoumaa2@illinois.edu). Received 21 December 2010 Accepted 13 April 2011 This journal is © 2011 The Royal Society 2896