Planar Cosserat elasticity of materials with holes and intrusions I Jasiuk Department of Materials Science and Mechanics, Michigan State University, E Lansing M148824-1226 USA M Ostoja-Starzewski Institute of Paper Science and Technology, 500 l Oth Street NW, Atlanta GA 30318-5794 USA Recently, Cherkaev, Lurie, and Milton (I 992) established an invariance of stress field in planar linear anisotropic elasticity under a specific shift in bulk and shear moduli; this is now known as the CLM theorem. Motivated by the importance of micropolar models in mechanics of media with micropolar structures, Ostaja-Starzewski and Jasiuk (1995) generalized the CLM theorem to planar micropolar elastic materials and considered inhomogeneous simply-connected materials. The present study addresses inhomogeneous, multiply-connected materials (with holes), which require global compatibility conditions involving Ces/Lro integrals, as well as multi-phase simply-connected materials, where the interface conditions need to be considered. Just as in the previous paper, both of these cases display a reduction in the parameter space. 1. INTRODUCTION The classical continuum mechanics is based on the assumption that the interaction between any two continuum particles across an elementary area lying within the body occurs solely through the force traction vector t (_= t: ). As a result, this theory lacks internal moment interactions~ and any intrinsic length scales, and hence, effectively, presents just a first-order approximation to a number of problems with microstructures. Effects of higher order are typically observed when one, or more, of the characteristic dimensions of the body decrease and become comparable to the typical material length scale(s) - such as the grain or crystal size - and as a result the microstructure gives rise to high local gra- dients. This happens, for example, in case of stress/strain concentrations in the vicinity of notches and internal defects, in mechanics of granular and multiphase media (e.g. soils, polymers, fluid suspensions), in mechanics of perforated plates, as well as in elastic vibrations of high frequency and short wavelength propagation comparable to microscale dimensions. First attempts to remove the restrictions of the classical continuum mechanics, in the context of elasticity theory, were due to Voigt (1887). Namely, he assumed that the inter- action between the continuum particles through a surface ele- ment dA occurs not only through a force vector t.dA , but 1 also through a moment vector m.dA , and consequently, the 1 conventional stress field becomes asymmetric and is accom- panied by an asymmetric couple-stres field. These ideas were fully developed at the beginning of this century by the broth- ers Eugene and Franqois Cosserat (1909) who constructed a fully consistent theory of a continuum, in which each point has six degrees of freedom of a rigid body, i.e. it is made of part of MECHANICS PAN-AMERICA 1995, edited by LA Godoy, SR Appl Mech Rev vol 48, no 11, part 2, November 1995 interconnected material particles, each capable of displace- ment u i and rotation ~i' which are, in general, independent functions of position and time. A new medium was thus described in which points acquired an orientation, i.e. a polar medium. The assumption of force transmission through the force traction t i and the couple (moment) m i leads, through the Euler-Cauchy principle, to two asymmetric tensors: force stress tensor cYij and couple stress tensor I.tij (e.g. Nowacki, 1986) = o..n. m i (1.1) ti J1 J = ktjinj The Cosserat theory remained practically unnoticed for half a century. This was likely due to the relative success at that time of the conventional continuum theories (classical elasticity and fluid mechanics), and to its generality (as a non- linear theory with finite motions and inelastic interactions) and its presentation as a unified theory incorporating mechan- ics, optics and electrodynamics. However, with many mod- ern technological developments in the post 2nd World War era - such as liquid crystals, porous and granular media, composites, complex engineering structures, complex fluids, etc. - there arose a need for more adequate continuum mod- els than the classical one. In the late fifties and early sixties the work of the Cosserat brothers was rediscovered and revived - it began to play an important role right at the base of continuum mechanics and thermodynamics that began then to undergo a dramatic devel- opment. As it soon turned out, the search for new models that would allow one to better describe a number of complex materials, resulted in theories close to, or identical to, the Cosserat model. In those early days the attention was initially focused on the simpler couple stress theory, or a (Cosserat) Idelsohn, PAA Laura, and DT Mook ASME Repdnt No AMP177 $96 $11 © 1995 American Society of Mechanical Engineers Downloaded 22 Oct 2009 to 130.126.179.16. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm