COMPLEX STRESS DISTRIBUTIONS IN ENGINEERING MATERIALS. 315 
this solution is at once seen to involve the ratio of the elastic constants, since o 
‘enters into the values of a,, a, determined by equations (21). 
These values can be simply expressed in terms of the force resultants X,, Y, 
applied to the corresp’ nding hole, For using (17) we have 
l—o \e (Fe) 1—"te | 
ou, = o_] \f [ae = F, 
Be ( -—o \ Ge 1+ f 
l—o , fo 1 — Nyx 
Dm =(, 9 — 1 Cy( +) = = =x 
My % (+ y, . L+n) Oo 
(22) 
9, These results are going to enable us, not only to obtain an upper I'mit to the 
divergences in the stresses in two circular rings of different materials under the same 
applied forces, owing to the divergence in the ratio of the elastic constants, but, 
what is even more important to correct definitely results obtained on a multiply- 
connected xylonite model and make them applicable to a material like steel, even 
when the mathematical solution cannot be attained. 
Let us take this latter application first. 
Suppose that our ‘ideal’ material is now taken to be xylonite, and that in a plate 
of xylonite the stiesses P, Q, S have been measured by the optical and transverse 
contraction methods described by Professor Coker [(7) (8) and (9)], so that we may 
consider P,, Q,, 8, as known (the suffixes having the same meaning as before). 
Suppose further that by a separate experiment with a xylonite model suitably cut 
and re-cemented, the stresses P,, Q,, 8,; P., Q,, S,; etc., corresponding to specified 
unit dislocations, have been similarly explored. As the only dislocations considered 
are translational, the experiment is very easily carried out in the majority of cases 
by cutting a straight channel perpendicular to the direction of translation required 
and bringing the edges of the cut together. 
Fie, 12. 
Now let the ‘actual’ material be any ordinary engineering material, e.g. steel, 
then the stresses in this material, due to the same force system already applied to 
the celluloid, are given by 
\ 
P=P,+4,P, +a,P,+ 
Q=Q,+4,Q,+4,Q,+ ... , ; (28) 
S=S8,+4,8,+4,S,+ f 
where a,, a,, a. . . . are given by equations (22) and similar equations. 
Now the total force resultants (per unit thickness) applied to the various 
boundaries are, in most problems to which this method can be applied, directly 
known, and the values of 7, 7), », for the materials concerned can be found without 
difficulty by direct experiment when they are not already known. Thus the a's are 
calculated without difficulty and the stresses in the steel plate at all points can be 
deduced. Accordingly the exploration of a xylonite model can, even in this case, be 
made to give complete information about the stresses in a plate of any material, 
even though the mathematical solution is beyond our present powers of analysis. 
10. We will now apply this method to find the magnitude of the corrections 
which have to be applied to the stress-system observed ina xylonite ring, the radii of 
the inner and outer boundaries being a and b. In order to do so it is necessary first 
of all to work out the stresses for the two elementary translational dislocations 
represented in fig. 12. 
1921 Z 
