Deformation and Motion
Based on physical considerations, the mathematical model of a micropolar continuum is introduced. Coordinates, base vectors, and shifters are presented in Art. 1.2. The concept of directors, fundamental to micropolar bodies, are discussed in Art. 1.3. The
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No. 23
CEMAL ERINGEN UNIVERSITY OF PRINCETO:'il
FOUNDATIONS OF MICROPOLAR THERMOELASTICITY
COURSE HELD AT THE DEPARTMENT
FOR MECHANICS OF DEFORMABLE BODIES JULY 1970
UDINE 1970
SPRINGER-VERLAG WIEN GMBH
ISBN 978-3-211-81142-9
ISBN 978-3-7091-2904-3 (eBook)
DOI 10.1007/978-3-7091-2904-3
Copyright
1970 by
Springer-Verlag Wien
Originally published by Springer Vienna in 1970
INTRODUCTION. The present work is devoted to the foundation of micropolar thermoelasticity. Essentially, it is intended for the development of the exact nonlinear theory. However the linear theory is produced as an approximation to the complete nonlinear theory. Micropolar continuum mechanics is a scientific discipline concerned with the mechanics of oriented bodies whose primitive elements (roughly speaking) consist of rigid particles. Contrasted to classical continuum mechanics, the material points of a micropolar continuum are endowed with intrinsic orientations and rotary inertia. These additional degreee cf freedom are believed to provide the proper physical mechanism to discuss and predict certain phenomena inherentZy due to the granuZar and moZecuZar nature of materials. In the hierarchy of micromorphic and other nonlocal continuum theories, micropolar mechanics is a sensible first step with its solid mathematical and physicaZ foundations and yet it is simple enough to permit serious mathematical work for the treatment of nontrivial boundary and initial value problems (in the linear theory). Applications of this theory are found and more are expected as the fields of composite materials, liquid crystals, granular solids etc. grow. WhiZe the ideas of an oriented continu-
4
Introduot ion
um oan be traoed aZl the way back to Bernoulli and Euler, in the 18th oentury, in oonneotion with their work on beam theories, to MaoCullog h [1839] , in oonneotion with his theory of optios, to Duhem [1893] in thermodyn amios,to Voigt [1887] in his work on orystallog raphy and to others at the end of the nineteenth oentury, the first systematio work on elastio solids, bars and plates was published by E. and F. Cosserat. The important monograph of the Cosserats ' [1909] was buried in the Literature nearly half a oentury until the topio was reopened and/or redisoovered reoently. Sinae already several expositian s exist on the historiaa l developme nt of this fieZd and more general theories of polar oontinua, of. Eringen [1967a] , [1968] , Stojanowia [ 1969] , we do not intend to traoe the history here. However a remark may be in order : It oan be said that one of the greatest oontribut ions of the Gosserats was the introduot ion if a new explioit kinematio s for a oontinuum with rigid direotors whioh is amenable to a simple but elegant interpreta tion of the motion. However, the Cosserats· ' work possesses two basio disadvanta ges : (a) It is soZely based on a variationa~ prinoiple and thus the resultant oonstitut ive equations are appZioabl e to elastioity only. (b) It Zaoks an expZioit form for the spin energy, furth
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