Abstract
Chapter 2 begins with the definition of the elastic potential from the strain energy and discusses the thermodynamic definition. Experimental observations concerning rubber elasticity are described followed by the explanation of models. Discussion of the cohesion energy allows classifying the various types of atomic bonds. Moduli and compliances in linear elasticity are described for anisotropic and then isotropic materials. Stability of the equilibrium is discussed and field equations are described and illustrated by the example of the propagation of plane sine waves. This is followed by the explanation of extremum theorems and an introduction to the finite elements method.
Chapter 2 continues with a discussion of homogenisation (estimating and bounding) for heterogeneous materials in linear elasticity, with explanations about the effective moduli and compliances, initial deformations, thermo-elasticity and the Voigt’s and Reuss’s bounds. A section is devoted to the problems of inclusions, ellipsoidal essentially.
Lastly, Chapter 2 is concerned with sharper bounds and improved estimates for the elastic moduli and compliances: Mori-Tanaka model and Hashin-Shtrikman bounds, self-consistent scheme. Finally an outline of the theory of elastic random media is given.
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Notes
- 1.
Augustin-Louis Cauchy (1789–1857) was a French mathematician.
- 2.
The definitions of the stress and strain tensors at finite strain are given in Annex 3 of volume III.
- 3.
Rudolf Clausius (1822–1888) was a German physicist.
- 4.
Pierre Duhem (1861–1916) was a French physicist.
- 5.
Gabrio Piola (1794–1850) was an Italian physicist.
- 6.
Gustav Kirchhoff (1824–1887) was a German physicist.
- 7.
Joseph-Louis Lagrange (1736–1813) was an Italian born, French mathematician.
- 8.
Georges Green (1793–1841) was a British mathematician.
- 9.
Adrien-Marie Legendre (1752–1833) was a French mathematician.
- 10.
Leopold Kronecker (1823–1891) was a German mathematician.
- 11.
The word ‘rubber’ is derived from the ability of natural rubber to remove (rub off) marks from paper, which was noted by Joseph Priestley, an English scientist (1733–1804), in 1770.
- 12.
Vulcanisation was discovered in 1839 by Goodyear and Hyward.
- 13.
John Douglas Ferry (1912–2002) was professor at the University of Wisconsin.
- 14.
Ludwig Boltzmann (1844–1906) was an Austrian physicist famous for his founding contributions in the fields of statistical mechanics and thermodynamics. The so-called “Boltzmann constant” was actually first introduced by the German physicist Max Planck (1858–1947), the founder of the quantum theory (Nobel Prize in Physics, 1918).
- 15.
Johannes-Diderick van der Waals (1837–1923) was a Dutch physicist.
- 16.
Max Born (1882–1970) was a German physicist.
- 17.
Thomas Young (1773–1829) was an English physician and scientist, who mastered eight languages.
- 18.
Siméon Poisson (1781–1840) was a French mathematician.
- 19.
Woldemar Voigt (1850–1919) was a German physicist.
- 20.
Gabriel Lamé (1795–1870) was a French mathematician.
- 21.
Émile Clapeyron (1799–1864) was a French physicist.
- 22.
Henri Navier (1785–1836) was a French physicist.
- 23.
Valery M. Levin is a Russian Professor of Mechanics at Petrozavodsk State University (Russia).
- 24.
Note that the above calculation could have been strongly shortened by the use of the fourth-order unit tensors J and K defined by J ijkl = (1/3)δ ij δ kl and K + J = I. They are such that c = 2μ K + 3 k J and s = (1/2 μ)K + (1/3 k)J. When acting on a symmetric second-order tensor such as ε ij , they yield directly K ijkl ε kl = e ij (deviator) and J ijkl ε kl = (1/3)ε kk δ ij (see Exercises of volume III for more applications).
- 25.
George Green (1793–1841) was a British mathematician.
- 26.
John Douglas Eshelby (1916–1988) was a British scientist.
- 27.
Zvi Hashin (1929–) is an Israeli physicist and engineer, Emeritus Professor at the Tel Aviv University. He derived the Hashin-Shtrikman bounds in 1963 in collaboration with his colleague Shmuel Shtrikman (1930–2003), a Belarusian-born physicist of the Weizmann Institute of Science, Rehovot, Israel, and of the Franklin Institute Laboratories, Philadelphia, USA, when he was a faculty member at the University of Pennsylvania, Philadelphia (USA).
- 28.
Albert Einstein (1879–1955) who is quoted here is the same German-born theoretical physicist as the universally known father of modern physics.
- 29.
Ekkehart Kröner (1919–2000) was a German physicist and a Professor at the University of Stuttgart (FRG).
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François, D., Pineau, A., Zaoui, A. (2012). Elastic Behaviour. In: Mechanical Behaviour of Materials. Solid Mechanics and Its Applications, vol 180. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-2546-1_2
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