Variational Principles and Applications in Finite Elastic Strain Theory

Citation

Levinson, Mark (1964) Variational Principles and Applications in Finite Elastic Strain Theory. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/PK58-HQ71. https://resolver.caltech.edu/CaltechETD:etd-09272002-145537

Abstract

The variational principles of finite elastostatic strain theory are presented in a unified manner for both compressible and incompressible bodies. Whereas the principle of stationary potential energy, a restricted case of the general principle of Hu and Washizu, is valid for any elastic deformation, it is found that the principle of stationary complementary energy is valid only for infinitesimal elastic strains. Consequently, Reissner's Theorem is the appropriate stationary principle to use in finite elastic strain theory when the complementary strain energy density is to be the argument function. The potential energy principle is applied to several problems dealing with the finite straining of a neo-Hookean material. All but one of these problems are concerned with plane strain deformations; the one other problem, in a spherical geometry, involves an unusual stability question. Approximate solutions are obtained for some mixed boundary value problems which are not amenable to the semi-inverse methods of solution frequently used in finite elastic strain theory. Another plane strain problem, requiring more detailed stress information than can be obtained from the potential energy principle, is studied approximately by means of Reissner's Theorem.

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