Byskov, Esben

Warren, William E.

English

    Aalborg UniversitetA Solution to Plane Problems of MicropolarelasticityWarren, William E.; Byskov, EsbenPublished in:First American Academy of Mechanics Conference 2008Publication date:2008Document VersionPublisher's PDF, also known as Version of recordLink to publication from Aalborg UniversityCitation for published version (APA):Warren, W. E., & Byskov, E. (2008). A Solution to Plane Problems of Micropolarelasticity. In First AmericanAcademy of Mechanics Conference 2008: June 17-20, 2008General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            ? Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            ? You may not further distribute the material or use it for any profit-making activity or commercial gain            ? You may freely distribute the URL identifying the publication in the public portal ?Take down policyIf you believe that this document breaches copyright please contact us at vbn@aub.aau.dk providing details, and we will remove access tothe work immediately and investigate your claim.Downloaded from vbn.aau.dk on: November 28, 2020A solution to plane problems of micropolarelasticityWilliam E. Warren∗ †ProfessorEsben Byskov∗ProfessorA SOLUTION TO PLANE PROBLEMS OF MICROPOLAR ELASTICITY Analysisof the mechanical properties of engineering materials with microstructure generally requiresmodification 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 insome materials presents a significantproblem in modeling these materials as continua in cer-tain situations. Of particular concern are those situations in which severe deformation occurswithin highly localized regions of the material where the size scale of the deformation regionis of the same order as that of the microstructure. One approach which recognizes specificmicrostructure is the theory of micropolar materials. In this model, a material point withinthe 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.Thismodel obviously encounters severe philosophical difficulties if considered as representative ofa 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 therelative rotations as well as the displacements of the two masses. Under these conditions, acharacteristic length is associated with the material, and the resulting stress and strain ten-sors are generally non-symmetric. In two-dimensional material models these microstructuresare often represented by hexagonal and equilateral trianglegeometries which exhibit three-fold symmetry in the plane. For two-dimensional linearly elastic micropolar materials, thisthree-fold symmetry condition requires that both the stresses and couplestresses be isotropicin the plane. In this work we obtain a general solution to the field equations of plane microp-olar elasticity in terms of three potential functions. Two of these functions are analytic, andthe third function satisfies the modified homogeneous Helmholtz equation. Representationsfor the complex displacement vector, the complex stress combinations, the complex couplestress, the rotation, and the resultant complex force on a contour are presented in termsof these potential functions. Solutions for the infinite plane with either a stress and couplestress free or displacement and rotation free circular boundary and loaded in the far field bya uniaxial stress have been obtained. These circular boundary problems do not provide largedisplacement gradients and the micropolar effects are quite small. We have also investigatedproblems in which the loading induces large localized displacement gradients for which themicropolar effects are significant. These include the classical crack problem and the half spaceloaded uniformly over a finite length of the boundary.∗Mech. Eng., UNM, 1401 Wells Dr. NE, Albuquerque, New Mexico 87112-6385†Corresponding Author; Contact Email: wewarren@osogrande.com170024

A Solution to Plane Problems of Micropolarelasticity

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A Solution to Plane Problems of Micropolarelasticity

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