146 PROF. E. G. COKER AND MR. K. C. CHAKKO: THE STRESS-STRAIN PROPERTIES 
the beam at the points where the colour in the specimen is neutralised is f 0 and the 
corresponding thickness is t 0 , we have the equivalent stress in the specimen . 
t 
Now if M is the bending moment in the beam, d is its depth, and y is the distance 
from the neutral axis, then 
fo = 
so that the equivalent stress 
Mo/ _ M y _ 12M y 
I fh tyP t 0 d s 
r __ /A 12M y 
J t td* ‘ 
Although as stated above the law of optical retardation is generally assumed to 
follow a linear law of stress difference, yet there is no apparent reason why it should 
not follow some other law, as for example a linear strain law or possibly contain terms 
involving squares of stress or strain. Some attempt has been made to find if the 
latter assumptions have any foundation, but if so the effects are within the limit of 
experimental error, and too small to be of any significance with the effect produced 
by a linear relation. 
As regards the question whether this relation should be expressed in terms of stress 
or strain, it may be pointed out that an attempt is made here to test this with 
materials under direct stress, and that the validity of the law for combined stresses 
and strains still remains for consideration (apart from lateral strains, which are 
presumed to have no effect beyond altering the length of the path in which retardation 
takes place), but as in this case, if the standard, not stressed beyond the elastic limit 
is compared with another in which this condition is passed the experiments do in fact 
provide a means of discrimination, since in the standard, stress and strain are 
proportional, but are not so in general for the tension member. Hence if the form 
of the law of optical effect is assumed in terms of stress it does not exclude the 
possibility of finding from the experimental evidence whether it should not be 
expressed in terms of strain. We may, therefore, without loss of generality, take as 
an assumption the usual relation that relative retardation R = C (P —Q) Us a con¬ 
venient expression where (P — Q) is the difference of principal stress = f, t is the 
thickness of the material and C is the stress-optical coefficient. 
Let C 0 be the stress-optical coefficient of the standard beam. Then R 0 = C 0< // 0 
for this beam. 
When this latter is used to neutralise the retardation R in the specimen, since 
R = R 0 , we have 
Qft = C 0 fot 0 
But t = t 0 , here and therefore 
c f= c 0 /;. 
