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Variational Principles and Energy Theorems

  • Danton Gutierrez-LeminiEmail author
Chapter
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Abstract

This chapter introduces the subject of the variation of a functional and develops variational principles of instantaneous type which are the equivalent of Castigliano’s theorems of elasticity for computing the generalized force associated with a generalized displacement and vice versa, by means of partial derivatives of the potential energy and the complementary potential energy functionals, respectively. A natural consequence of the variational principle of instantaneous type is that the constitutive potentials of viscoelastic materials are not unique. Any dissipative term can be added to them without changing the stress strain law. The viscoelastic versions of the unit load theorem of elasticity, and the theorems of Betti and Maxwell for elastic bodies, are also developed in detail.

Keywords

Variation Variational Functional Potential Stationary Admissible Field Castigliano Unit load Reciprocal 

References

  1. 1.
    Y.C. Fung, Foundations of Solid Mechanics, Prentice Hall, Inc., p 429–433 (1965)Google Scholar
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    A. Ghali, A.M. Neville, Structural Analysis. A Unified Classical and Matrix Approach, International Textbook Co., p. 90–110 (1972)Google Scholar
  3. 3.
    R.M. Christensen, Theory of Viscoelasticity, 2nd Ed., Dover, p. 3–9 (1982)Google Scholar
  4. 4.
    T. Oden, Mechanics of Elastic Structures, McGraw-Hill, pp. 76–78, 89–93 (1967)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Special Products DivisionOil States Industries, Inc.ArlingtonUSA

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