# A Solution to Plane Problems of Micropolarelasticity

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## Abstract

A solution to plane problems of micropolar elasticity William E. Warren∗ † Professor Esben Byskov∗ Professor A SOLUTION TO PLANE PROBLEMS OF MICROPOLAR ELASTICITY Analysis of the mechanical properties of engineering materials with microstructure generally requires modification of the concept of a simple material. A number of sophisticated averaging tech- niques have been developed to model these materials with microstructure within the frame- work of classical continuum mechanics. However, the nature of a discrete microstructure in some materials presents a significantproblem in modeling these materials as continua in cer- tain situations. Of particular concern are those situations in which severe deformation occurs within highly localized regions of the material where the size scale of the deformation region is of the same order as that of the microstructure. One approach which recognizes specific microstructure is the theory of micropolar materials. In this model, a material point within the body undergoes the usual displacement from its original position, and in addition, the ma- terial point is assumed to undergo a rotation which is independent of the displacement.This model obviously encounters severe philosophical difficulties if considered as representative of a continuum. It does, however, reflect the mechanics of a discrete structure where, for exam- ple, the forces and moments in a beam connecting two masses requires information about the relative rotations as well as the displacements of the two masses. Under these conditions, a characteristic length is associated with the material, and the resulting stress and strain ten- sors are generally non-symmetric. In two-dimensional material models these microstructures are often represented by hexagonal and equilateral trianglegeometries which exhibit three- fold symmetry in the plane. For two-dimensional linearly elastic micropolar materials, this three-fold symmetry condition requires tha

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