554 
SGc 
area Aoe Let 1 be the length of this portion, r the radius and a the cross-sectional area when 
it is subjected to a longitudinal compressive stress P, the cylindrica) surface being free from stress. 
tf o be Poisson's ratio for the material of the bar, then 
P r 
ten ieee x P 
=; rot ito. 
° 
The quantities P/E, OP/E and similar quantities are strains which we assume to be sufficiently 
small to allow squares and higher powers to be neglected. It follows that 
2 
f 
5 ial) ahaa SiH lise iste Meron d hee (u.5a) 
QO 
and, from the conservation of mass, 
ee Ge) eee beedapssonregsediany Le (4.5) 
Po4o Ue E 
Next Suppose that the compressive stress P is removed and that a uniform hydrostatic pressure, 
Py is applied to the portion of the bar, and that its length, radius and cross-sectional area become 
1 
1" rs and Ay respectively. we now have 
) r P 
1 1 1 
Ss mmey Hg) fae ES (Gt g7) 
WD a E 
so that 
Ay r2 
Jos + Vere Page 2) ese nase ewe ae (4, 6a) 
Ay 0 —_— 
€ 
and 
PA 
shal 1 P 
MOM =e hy Uti aSo)) Meletd Ne isle, sieciet clere (4.6) 
Poh Ty E 
From equation (4.3) 
rtm ay PA 
fc ee ee 
uU- ou PsA, Scone st 
from equations (4.5) and (4.6). 
Neglecting squares and higher orders of the smal) quantities P/E... u,/U, u/U, equation 
(4.7) becomes 
ea gia [p-P, (y - yon) mets tice Le SCLLLONG (4.8) 
From equation (4,4) 
PA(U = u) E asd auone: E(wSO) 
from sscees 
