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Erschienen in:

2014 | OriginalPaper | Buchkapitel

5. Structural Mechanics

verfasst von : Danton Gutierrez-Lemini

Erschienen in: Engineering Viscoelasticity

Verlag: Springer US

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Abstract

This chapter is devoted to structural mechanics, developing the theories of bending, torsion, and buckling of straight bars, and presenting a detailed account of vibration of single-degree-of-freedom viscoelastic systems, including vibration isolation. A balanced treatment is given to stress–strain equations of integral and differential types, and to stress–strain relations in complex-variable form, which are applicable to steady-state response to oscillatory loading. All equations in this chapter are developed from first principles, without presuming previous knowledge of the subject matter being presented. This approach is followed for two reasons: first, because it is necessary for readers without a formal training in mechanics of materials; and secondly, because it provides the reader—even one with formal training in classical engineering—with a method to follow when the use of popular shortcuts, like the integral transform techniques, might be questionable or unclear.

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Vorheriges Kapitel Constitutive Equations for Steady-State Oscillations

Nächstes Kapitel Temperature Effects

That is, such that the external stimuli vary sufficiently slowly that no significant inertial effects develop.

The mechanical elements that may act on a bar’s cross section are two bending moments, one torsional moment, one axial force, and two shear forces.

In this chapter, to avoid confusion, the letter M is used to denote the moment of the external forces, while E will denote relaxation modulus. As in previous chapters, C will still denote compliance.

Generalization to beam-columns, which include axial loads, is straightforward and is left as an exercise.

Note that the same result may be reached by taking the kinematic relation (5.3) into the stress–strain constitutive equation σ(t) = E(t − τ)*dε(τ).

Here \( E^{*} \left( {j\omega } \right) \equiv \left( {j\omega } \right)\mathop \smallint _{0}^{\infty } e^{ - j\omega t} E\left( t \right)dt \equiv \left( {j\omega } \right)\mathcal{F}[E \left( t \right)] \); where \( \mathcal{F}[E \left( t \right)] \) is the Fourier transform of E(t).

As shown in Chap. 4, \( Q\left( {j\omega } \right)/P\left( {j\omega } \right) = \left( {j\omega } \right) \cdot \mathcal{F}[E \left( t \right)] \equiv E^{*} \left( {j\omega } \right) \).

Note that, just as \( E^{*} \left( {j\omega t} \right) \equiv (j\omega ) \cdot \mathcal{F}[E\left( t \right)] \), for the uniaxial modulus in tension, the complex shear modulus is defined by \( G^{*} \left( {j\omega t} \right) \equiv (j\omega ) \cdot \mathcal{F}[G\left( t \right)] \).

The Carson transform of a function is the transform-variable multiplied transform of the function.

This is true for all materials, irrespective of their constitution.

In general, because the steady-state viscoelastic input and response are out of phase, whenever the complex-controlled variable is, say, c ^* = c _o e ^jωt, the corresponding response variable is of the form r ^* = r _o e ^j(ωt+δ) where the sign of the phase angle is positive if the response is of stress type, and negative, if it is of strain type.

E. Volterra, J.H. Gaines, Advanced Strength of Materials (Prentice-Hall, Inc., Englewood Cliffs, 1971), pp 257–268

J.T. Oden, Mechanics of Elastic Structures (McGraw-Hill Book Co., New York, 1967), pp 30–33

A. Chajes, Principles of Structural Stability Theory (Prentice-Hall, Englewood Cliffs, 1974), pp 1–3

R. M. Christensen, Theory of viscoelasticity, 2nd edn. (Dover, New York, 1982), pp. 21–26

A.N. Gent, Engineering with Rubber (Hanser, Munich 2001), pp. 73–87

Titel: Structural Mechanics
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_5

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