COMPLEX STRESS DISTRIBUTIONS IN ENGINEERING MATERIALS. 307 
appears, however, unaware that the unique determination of the stress function, 
independently of the elastic constants, is subject to exceptions. 
The following sections give a short account of the theory of such dislocations, in 
a plane, on lines somewhat more direct than those followed by Michell and Volterra. 
One important result obtained by this method of attack is to show that the correc- 
tions to the stresses under a given system of loads, when we pass from one material 
to another, can be obtained directly from exploration of the stresses in one kind of 
material only, by a subsidiary experiment with a dislocated plate, provided Poisson’s 
ratio is known for both materials; this correction, for any given shape of plate and 
appliec force system, can thus be determined experimentally by optical means, even 
when a mathematical solution is not available. 
It seemed also of interest to determine, for a shape of plate which allows of 
mathematical solution, an upper limit to the divergences to be expected in the 
stresses, when we pass from one material to another. This has been done here for a 
ring-shaped plate; the final results are simple in form and give indications which 
should be valuable to the engineer. In particular they show that if the width of the 
ring be moderate in comparison with its radius, the corrections, in practice, will be 
very small, at any rate at all points where the stresses are of any magnitude. 
4, In what follows we shall have to consider functions of which the derived 
functions of a certain order are one-valued, whilst the functions themselves and 
their derived functions of lower order are many-valued. 
Let ¢ be a function of wz, y, and let its differential coefficients be detined in a non- 
ambiguous manner all over the area contained by a closed circuit APBQA (fig. 10), 
Pp 
A A 
Q 
dx 
Fig. 10. 
which can be made to shrink to a point without passing outside the material. Such 
a circuit is said to be reducible. 
Then if (), be the value of @ selected at A at the beginning of the circuit 
and (), the value reached at A after describing the circuit 
($)2—(), = [" uh ds taken round the circuit. 
JA O8 
(A (do bp ) 
= .du+ — dy). 
I, bx at By ay 
But since eg Bs have been defined in a non-ambiguous manner, then, in the figure 
aw by 
9) 7 [ Ag (2) ] 
— Axvp — div = du = — r 
(3 p ial (3). Ge = ci Q dx/p 
(bearing in mind the sense of describing the contour) 
were dw, dy are now positive throughout. 
