
( 473) 
XXIII.—On the Stresses due to Compound Strains. By Professor C. Niven. 
Communicated by Professor Tarr. 
(Received 22d January 1874.—Read 16th February 1874.) 
§ 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 Caucny, 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. Dr Sr Venant and Bovssinesg, 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 Caucuy’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 discus- 
sion of this problem. These authors have also failed to see in what way the 
remaining part of the potential depends on the original strain. 
In the present paper the treatment of the matter is grounded solely on the 
laws according to which one set of distortions may be superposed on another. 
The resultant strains, it is shown, differ from the primary by linear functions of 
the secondary ones, whether the latter be small or large. The general expres- 
sion for the potential energy is thus found independently of any hypothesis, 
and so far coincides with the result of M. Bousstnesa. In the further develop- 
ment of the subject, I have confined myself to the case where the potential 
due to the primary strain is a quadratic function of the corresponding strains, 
and also where the substance was originally isotropic.* In this case, it appears 
that the increase of the potential energy, so far as it involves the primary 
strain, depends only on six quantities called quasi-strains. The primary stresses, 
each multiplied by the dilated unit-volume, also depend only on these six 
functions. 
* See, however, the note added 16th February 1876. 
VOL. XXVII. PART IV. 6 kK 
