220 
MR. L. N. r4. FILOX OX THE ELASTIC EQTMLTBrJT'>f OF 
But the solution, when the ends are compelled to expand by a given quantity a,-,, 
is easily dedncible from that given for non-expanding ends. Thus, let Wi, i(\ be the 
values of n and iv in the case worked out, when Q = 1, and let us widte 
w = P 
2/a(3X + '2 fi ) 
IV = P 
\ yLt (3X + 2/x)j 
+ • 
Then this will satisfy all the conditions, provided 
and 
Tlierefoi'e 
X 
^ 6X + 4yU ^ 
P + P Q. 
n X 
(6\ -p 4g) 
2 (X + g) “'OP'px , , X \ 
X 
giving the solution under a given mean pressure Q, which produces a flow a,^ of a 
lead plate, and thereby constrains the ends of the test piece to expand by that 
amount. 
'flie principal stress at the perimeter of the section is now 
P + (I-680) n = 1-G8f) Q - -080 P, 
and if P, be., l)e made large enough, this can be made much smaller than Q. It 
))egins to be smaller than Q as soon as the expansion induced by the flow of the lead 
is greater than the natural expansion of the stone under uniform pressure. 
On tlie other hand, the principal stress at the centre of the plane ends is 
P + -080 Pt = -686 Q + -314 P, 
and this again may be made great by making large. 
The principal stress-dlfl'erences are : 
at the perimeter t’G86 Q — •G8G P 
at tlie centre P — •211 P = — •211 Q -b 1^211 P. 
Hence we see that wliatever theory of failure we adopt, if the ends are forced to 
expand, so that P > Q, the material first becomes plastic (or else breaks) at the centre 
of the cross-section, the strength of the test-piece diminishing as P Increases, b\it 
liavlng no definite value. Tliat some such thing as this does really occur in practice 
is very well shown bv the results published by Unwin (‘ The Testing of ISlaterials of 
