Micropolar Media I. [antikvár]

Inl.J. EngngSci. Vol. 9. pp. 271-305. Pergamon Press 1971. Printed in Great Britain MICROPOLAR MEDIA - I THE CLASSICAL THEORYt K C. B. KAFADAR and A. CEMAL ERINGEN Princeton University, Princeton, N.J. 08540, U.S.A. Abstract-The present paper (Part I) is concerned with the nonlinear theory of micropolar média. The kinematics, balance laws, and constitutive theory are examined and utilized to develop the nonlinear theory of micropolar elasticity through a function basis. The fundamental problems solved include (i) motions independent of response functions, (ii) micropolar acceleration waves, and (iii) constrained microrotation and its application to rubber bonded bárium ferrites. In Part II we develop the relativistic (special) theory of micropolar média. INTRODUCTION This work is concerned with the foundations of the nonlinear theory of polar média. It is divided into two parts: Part I (the present paper) discusses the classical theory and Part II (the companion paper) deals with the relativistic (special) theory. In 1909, E. and F. Cosserat[4], published an important monograph that-while neglected for nearly a half century - first presented in a generál manner many of the concepts acknowledged requisite for polar média. While no attempt will be made toward giving the complete historical development, the Cosserats' work directly or indirectly spawned such diverse investigations as Günther[19], Eringen and Suhubi [10, 12], Eringen[ll, 14, 16], Toupin[30], and Green and Rivlin[17], among many others. Perhaps one of the Cosserats' greatest contributions was a new explicit rendering of kinematics peculiar to média that can support couples. Their concept of a rigid triad was amenable to a simple and beautiful geometrical interpretation of the motion. Yet the Cosserats' work had two disadvantages: (1) it was solely based on a variational principle and thus the resultant constitutive equations were applicable only to nondissipative média; (2) it lacked an explicit form for the kinetic energy and the so-called spin density (the dynamic terms, being defined by the Hamiltonian (E. and F. Cosserat [4], equations at the end of section 63) are suíficiently vague to be of any utility). This article relies heavily on the papers by Eringen in this area(what is occasionally called, by historical tradition, in this paper 'polar' is referred to as 'micropolar' by Eringen and is a special case of Eringen's more generál 'micromorphic' theory). The notation used throughout closely follows that of Eringen [9], and Truesdell and Toupin [3 I], It is implicitly assumed that the reader is familiar with contemporary continuum mechanics. The present investigation focuses on the nönrelativistic theory of polar média. Apart from its own interest, this provides the framework for the relativistic theory (companion paper). The first section is akinematical study of the 'micromotion' based on a decomposition involving the micromotion's angle and axis of rotation. The primitive physical quantities and the balance laws they obey having been stated (sections two and three), the fourth section introduces the concept of objectivity into tThis work was supported by the Office of Naval Research. 271