COMPLEX STRESS DISTRIBUTIONS IN ENGINEERING MATERIALS, 511 
7. Let us now consider in what way the elastic constants are introduced into 
the solution. 
Imagine a plate of the same size and shape as the one considered, but made 
: of some ideal material whose elastic constants have fixed numerical values. Then, 
if the given forces are applied to this ideal plate, there will necessarily on physical 
grounds be a solution which will lead to acyclic displacements Uy, Vy. Let E, be 
the value of E and y, the value of ~ corresponding to this solution. 
Then, as before, 
oR ay 
° ae ee (|) 0 
| Hes gt eg, ) 
3K 3y, | (15) 
Day peop ig ARO 
MoV by #( %o bc } 
u,, 7, referring to the ‘ideal’ material, so that these are fixed numbers. 
| Now, if the plate is simply connected, every circuit drawn on it is reducible, 
_ hence E, and y, are necessarily acyclic. ‘ 
If we now consider the same plate, made of any actval material, and write down 
_ the displacements 
2uU = — dE, + (1-9) Bw, 
ox by 
bead ; i : (16) 
H 5B By 
uN = ts CLS oy 
[o By Se ee | 
These displacements will clearly lead to the same stresses as before—since the 
stresses depend on E only, and E is here K,. Also since E, and y, are both acyclic 
_soare Uand V. We have therefore the solution required. 
Thus, for any material, E = K,, and so is independent of the elastic constants. 
4 We have now to inquire how this is modified when the plate is multiply 
connected, there being no dislocations. This requires that U and V shall be acyclic. 
First consider the case where the forces applied to: each boundary separately 
reduce toa couple. This requires (§5) that fat 5 are acyclic. Then, consider- 
. y 
ing the plate of ideal material 
; 
J 
(=a) Cy (FH) = 2u,cy (OD + Cy (FE) = 0, 
oy or 
5 ‘SE 
(1-0) Cy (3) = Qu, Cy (V,) + Cy ( ) =) 
Hence sus and ve are again acyclic, equations (16), therefore, lead to acyclic 
|; 
y xv 
t values of U and V for the actual plate whatever o may be, and we can take E=E, 
and thus again the stresses are independent of the elastic constants. 
If, however, the forces applied to each boundary do not reduce separately to 
a couple, then ae and = are cyclic, and in order to make the displacements 
ec Y 
U,, V, acyclic, it is necessary that 
o Vo acy J 
or (zat) = ex) 
7 or) = -ro0(%) 
Hence, for the actual pasted when o+0, 
cy (=) rma = 003 ( os 
Cy (iy) + (1-0) Cy( 5). 
