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Erschienen in:

2014 | OriginalPaper | Buchkapitel

11. Variational Principles and Energy Theorems

verfasst von : Danton Gutierrez-Lemini

Erschienen in: Engineering Viscoelasticity

Verlag: Springer US

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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.

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Vorheriges Kapitel Wave Propagation

Nächstes Kapitel Errata to: Engineering Viscoelasticity

Viscoelastic variational principles concerned with variations of the histories of the arguments will be included in subsequent editions of the text.

In keeping with standard practice, in this chapter only, we use the Greek letter, δ, to denote variation of a function or functional; and as such, is not to be mistaken for the unit impulse or Dirac Delta function, nor for the Kronecker delta, used to represent the unit tensor.

As in previous chapters, the subscript g stands for “glassy”, to indicate the value of the corresponding material property function at t = 0.

The requirement of continuity up to the third order derivatives of the displacement field stems from the fact that the equations of compatibility involve second derivatives of the strains, which are defined in terms of the first derivatives of the displacement field.

The negative sign to define Y _to is only used for mathematical convenience.

The term generalized force is used to denote either a concentrated force or a concentrated moment, while the term generalized displacement denotes either a linear or an angular displacement. A generalized force and a generalized displacement are work-conjugate if the work done by the former acting on the later can be correctly calculated from their product.

For simplicity of exposition, the prime notation used to distinguish statically admissible fields is dropped.

This is so, because mixed terms involve integrals of the form: $ \int\limits_{A} {ydA;} $ which are identically zero by the assumption that the reference axes are centroidal.

The sense of the generalized unit load is arbitrarily chosen; but if the computed deflection is negative, its sense will be opposite that assumed for the generalized unit load.

A centroidal and principal coordinate system is located at the center of area –or centroid– of a cross section, with its axes coinciding with the cross section’s principal axes of inertia. In such a system, the following relations hold:

$$ \int\limits_{A} {xdA = } \int\limits_{A} {ydA = } \int\limits_{A} {rdA = } 0;I_{{\overline{xy} }} \equiv \int\limits_{A} {xydA = } 0;I_{{\overline{x} }} \equiv \int\limits_{A} {y^{2} dA} ;I_{{\overline{y} }} \equiv \int\limits_{A} {x^{2} dA} ;J_{{\overline{z} }} \equiv \int\limits_{A} {r^{2} dA} $$

Y.C. Fung, Foundations of Solid Mechanics, Prentice Hall, Inc., p 429–433 (1965)

A. Ghali, A.M. Neville, Structural Analysis. A Unified Classical and Matrix Approach, International Textbook Co., p. 90–110 (1972)

R.M. Christensen, Theory of Viscoelasticity, 2nd Ed., Dover, p. 3–9 (1982)

T. Oden, Mechanics of Elastic Structures, McGraw-Hill, pp. 76–78, 89–93 (1967)

Titel: Variational Principles and Energy Theorems
verfasst von: Danton Gutierrez-Lemini
Verlag: Springer US
Buch: Engineering Viscoelasticity
Print ISBN: 978-1-4614-8138-6

Electronic ISBN: 978-1-4614-8139-3

Copyright-Jahr: 2014
DOI: https://doi.org/10.1007/978-1-4614-8139-3_11

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