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Kelvin and the Early History of Seismic Anisotropy*)

By
Klaus Helbig
Klaus Helbig
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Published:
January 01, 1996

Abstract

In April 1856 Lord Kelvin (then still William Thomson) read a paper before the Royal Society, in which he outlined a description of the elastic tensor entirely based on the inherent symmetry of the material itself, i.e., without reference to any external coordinate system. The key concepts were the “perfect concurrence” (parallelity) and the orthogonality of stresses and strains, i.e., concepts known for vectors. This can be seen as a mapping of the space of tensors of rank 2 on a six-dimensional vector space. At the time of the presentation, the terms ‘tensor’ and ‘vector’ were not yet known, so that Kelvin had to develop his own terminology.

For this mapping to be meaningful, it must preserve the norm, which Kelvin deduced from the invariance of the elastic energy. This requires that the tensor components are written as components of six-vectors in a way that differs from the conventional ‘contracted notation’. In this notation, the elastic tensor becomes a second rank tensor in six-dimensional space (with components that differ from the elements of the conventional 6x6 matrix). The (mutually orthogonal) eigenvectors of the elastic tensor provide a reference system in which the elastic tensor is diagonalized. The decomposition of the 21 independent components of the elastic tensor into the six eigenvalues (eigenstiffnesses) and the 15 independent parameters of the eigenstrains provide the basis for the coordinate-free description of the elastic tensor: the six eigenstiffnesses encode the properties related to the magnitude of stresses and strains, twelve parameters of the eigenstrains encode the geometric relations of stresses and strains (among others the symmetry properties of the elastic tensor), and three of the parameters of the eigenstrains describe the orientation of the system of eigenstrains with respect to the (arbitrary) external reference system.

Though published in the Philosophical Transactions of 1856 and reprinted as an appendix to the entry «Elasticity» in the sixth edition of the Encyclopaedia Britannica (1878, vol. 7), the article seems to have made no impression on Kelvin’s contemporaries. The only reference known is a critical discussion in Todhunter and Pearson’s History of the Theory of Elasticity … of 1893, which shows that the writer was aware of but did not understand – Kelvin’s ideas. It has taken over hundred years for science to emulate Kelvin.

The concepts proposed by Kelvin (by now independently proposed by several contemporary authors) allow to handle many problems of the theory of elasticity more concisely (and more elegantly) than with the conventional formalism.

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Contents

Seismic Anisotropy

Erling Fjær
Erling Fjær
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Rune M. Holt
Rune M. Holt
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Jaswant S. Rathore
Jaswant S. Rathore
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Society of Exploration Geophysicists
ISBN electronic:
9781560802693
Publication date:
January 01, 1996

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