Abstract
Experimental demonstration of viscoelastic behaviour includes creep and relaxation tests, recovery tests and dynamic mechanical analysis (DMA). Formulation of linear viscoelastic functions are given and the Boltzman superposition principle explained. The Laplace-Carson transform allows simplifying the one-dimensional constitutive behaviour in non-ageing linear viscoelasticity. This leads to representation with the use of partial differential equations. Spectral representation is a generalisation. Thus, DMA can be performed. It is explained how to check the linearity of the behaviour.
Polymers are an important case of viscoelastic materials. This behaviour is linked to the conformation of chains in amorphous polymers. Temperature effects result: glass transition temperature, time-temperature superposition principle. The relaxation mechanisms and viscoelastic behaviour are discussed. The glass transition and the β transition are related to the structure of polymers. Physical ageing is discussed as well as the case of semi-crystalline polymers.
Another manifestation of viscoelasticity is internal friction in metals. There is a distinction between relaxation peaks and hysteretic behaviour. The point defect relaxations include the Snoek and the Zener effects. The Bordoni, the Hasiguti, the Snoek-Köster, the damping background at elevated temperatures are dislocation induced relaxation phenomena. Grain boundaries are also sources of relaxation internal friction. At higher amplitudes, dislocations produce hysteretic damping. Damping can be very high when due to phase transformations.
The 3-D formulation of constitutive equations is then given. The correspondence theorem allows structural design. Finally, the analysis of the overall behaviour of heterogeneous materials through the estimation of the effective creep moduli and compliances is discussed.
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Notes
- 1.
Oliver Heaviside (1850–1925) was a British physicist.
- 2.
Ludwig Boltzmann (1844–1906) was an Austrian physicist famous for his founding contributions in the fields of statistical mechanics and thermodynamics.
- 3.
Thomas Joannes Stieljes (1856–1894) was a Dutch mathematician.
- 4.
Pierre-Simon de Laplace (1749–1827) was a French mathematician and astronomer.
- 5.
John Renshaw Carson (1886–1940) was an American mathematician and electrical engineer.
- 6.
Hjalmar Mellin (1854–1933) was a Finnish mathematician.
- 7.
Thomas John l’Anson Bromwich (1875–1929) was an English mathematician.
- 8.
Augustin-Louis Cauchy (1789–1857) was a French mathematician.
- 9.
Gaspard Clair François Marie Riche de Prony (1755–1839) was a French mathematician and engineer.
- 10.
James Clerk Maxwell (1831–1879) was a Scottish physicist and mathematician.
- 11.
Willian Thomson, 1st Baron Kelvin (1824–1907), was a British mathematical physicist and engineer.
- 12.
Woldemar Voigt (1850–1919) was a German physicist.
- 13.
Clarence Melvin Zener (1905–1993) was an American physicist.
- 14.
Johannes (Jan) Martinus Burgers (1895–1981) was a Dutch physicist (see also Chap. 3).
- 15.
Kenneth Stewart Cole (1900–1984) was an American biophysicist. Robert H. Cole was his younger brother.
- 16.
Jean Baptiste Joseph Fourier (1768–1830) was a French mathematician.
- 17.
The book of Blanter et al. (2007) was of great help in writing this subsection.
- 18.
Peter Joseph Wilhem Debye (1884–1966) was a Dutch physicist and chemist who won the Nobel Prize in chemistry in 1936).
- 19.
Maurice Anthony Biot (1905–1985) was a Belgian-American physicist.
- 20.
Lars Onsager (1903–1976) was a Norwegian-born American physical chemist and theoretical physicist, winner of the 1968 Nobel Prize in Chemistry.
- 21.
Note that when the considered function f(t) is a relaxation function, according to (5.42), this approach reduces to the approximation of its continuous relaxation spectrum by a discrete one.
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Acknowledgments
The authors are particularly grateful to Professor Jacques Verdu (École nationale supérieure des Arts et Métiers, Paris) for his help in writing the entire subsection devoted to polymers, as well as part of annex 1.
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François, D., Pineau, A., Zaoui, A. (2012). Viscoelasticity. In: Mechanical Behaviour of Materials. Solid Mechanics and Its Applications, vol 180. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-2546-1_5
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