Eigenvalue problems associated with Korn's inequalities in the theory of elasticity

Applied Mechanics

Horgan, Cornelius Oliver ; Knowles, James K.

University:California Institute of Technology

Department:Engineering and Applied Science

关键词: Applied Mechanics;   

Others  :  https://thesis.library.caltech.edu/9082/1/Horgan_co_1970.pdf

美国|英语

来源: Caltech THESIS

【 摘 要 】

Interest in the possible applications of a priori inequalities in linear elasticity theory motivated the present investigation. Korn'sinequality under various side conditions is considered, with emphasison the Korn's constant. In the "second case" of Korn's inequality, avariational approach leads to an eigenvalue problem; it is shown that,for simply-connected two-dimensional regions, the problem of determining the spectrum of this eigenvalue problem is equivalent to findingthe values of Poisson's ratio for which the displacement boundary-valueproblem of linear homogeneous isotropic elastostatics has a non-unique solution.

Previous work on the uniqueness and non-uniqueness issue forthe latter problem is examined and the results applied to the spectrumof the Korn eigenvalue problem. In this way, further information onthe Korn constant for general regions is obtained.

A generalization of the "main case" of Korn's inequality is introducedand the associated eigenvalue problem is a gain related to thedisplacement boundary-value problem of linear elastostatics in twodimensions.