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XXIII.—On the Stresses due to Compound Strains

Published online by Cambridge University Press:  17 January 2013

C. Niven
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Extract

§ 1. The general problem in elasticity, as usually presented for solution, supposes the elastic substance to pass initially from a state without strain. But important cases exist where it must be conceived to start from a state already under considerable stress. When this is the case, the constitution of the solid undergoes great change, as is shown by the fact that strained glass loses its isotropic property, and becomes doubly refractive. This subject was long ago attacked by Cauchy, who, by means of the theory of molecular actions, deduced the existence, in the expressions for the stresses due to the secondary strains, of terms proportional to the initial or primary stresses. The problem has been since discussed by MM. De St Venant and Boussinesq, who have applied to it Green's expression for the energy stored up during the strain. But the question, in their hands, still retains traces of Cauchy's hypothetical element, inasmuch as their expression for the potential energy was deduced by means of the molecular theory. M. De St Venant even considers it a strong argument for the truth of the latter, that it is indispensable in the discussion of this problem. These authors have also failed to see in what way the remaining part of the potential depends on the original strain.

Type
Research Article
Information
Earth and Environmental Science Transactions of The Royal Society of Edinburgh , Volume 27 , Issue 4 , 1876 , pp. 473 - 491
Copyright
Copyright © Royal Society of Edinburgh 1876

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References

page 473 note * See, however, the note added 16th February 1876.

page 474 note * If the solid be “ incompressible ” for small strains, and if the primary strains be small, the equation giving the velocity of transmission of a plane wave is similar in form to that given by Fresnel in his Theory of Double Refraction.

page 489 note * It is perhaps worth noting that there is still a fourth invariant, which depends on the systems (AB … F), (a,b…f), namely, , but it does not present itself in these investigations. It may be observed, however, that it is not really independent of the nine magnitudes . For every invariant relating to two systems of strains, or to a system of strains and one of quasi-strains, can depend on only nine elements,—the six principal strains, and the three magnitudes which determine one set of principal strain-axes with regard to the other. In fact, referring one set of strains to their principal axes, we see that the ten invariants involve only nine independent quantities.

This is a special case of the more general theorem that n sets of magnitudes cogredient with strains give rise to n(2n+ 1) invariants apparently independent, but of which only 6n—3 are actually independent, the remaining 2n 2 — 5n + 3 being functions of these.