Treatise on the Mathematical Theory of Elasticity, Vol. 2 (Classic Reprint)

In the first volume of this work we have seen how the Theory of Elasticity took its rise in an enquiry of Galilei sconcerning the resistance of a beam to rupture by flexure; how the methods pursued in the 18th century in endeavours to answer this question led by a natural sequence to the discovery of the general equations of Elasticity; how this discovery was the prelude to a series of brilliant analytical researches concerned with problems of the most general character; and how, at the same time, it made possible an attempt to answer from mechanical principles questions of the deepest physical significance and the highest practical importance. The discussion of the number and meaning of the elastic constants threw light on the most recondite problems of intermolecular force ;the laws of wave-propagation in solids, both by agreement and disagreement with optical experiments, illustrated the nature of the luminiferous medium; the theories of the torsion and flexure of beams supplied the engineer with valuable working formulae. We saw that for practical application of these formulae it was necessary always that the length of the beam should be great compared with its thickness or that the linear dimensions of the body concerned should be of different orders of magnitude. We have now to consider in greater detail the special problems that arise when some linear diniensions of a body are small in comparison with others. These problems include the bending of rods, on which depends a large part of the theory of structures, the vibrations of bars, leading to theories of resilience and impact, the deformation of thin plates and its developments in regard to the vibrations of bells. Further, as we shall see hereafter, the series of difficult questions which turn upon elastic stability belong to the same class of problems. L. II.
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