PLANE STRAIN IN BIPOLAR CO-ORDINATES. 
269 
If the displacements in the directions normal to the curves a and /3 constant are 
u, v respectively, the strains are given by* 
; BU dll 
e a * = /o- v 
0 
r a 
, or o// 
^ = A —~u —■ 
op da 
0 
«-/» = — (AV) + 3-5 (A-W), 
na op 
and the corresponding components of stress by 
nrx — \ (e aa +e^) +2 /xe aa , 
— A (c aa + c^g) + 2fj.epp, 
a/3 = /X^a/3- 
(4) 
These stresses may be derived from a stress-function, so that in rectangular 
co-ordinates 
_ ^2 .—- 02 
~) „ 
Ox 
XX 
d\ ^ S 2 X ^ 0 2 x 
xy = - ' W = T2 
oy ox oy 
Transforming these equations to curvilinear co-ordinates we obtain 
= *4(*m)- a !t!M 
p \ 0/3/ 0a 0a 
0A 0Y 
aa 
w = h±(}M-hfJ* j. 
oa oa/ dp dp 
^2 
(5) 
o 2 (A x ) . , 0 2 A 
oa op oa op 
We will usually find it convenient to deal with Ax instead of x itself, and in our 
particular co-ordinates these equations become 
02 
acta. = \ (cosh a— cos /3) —5 — sinh a — — sin /3 ~ + cosh a l (Ax). 
{ op da op ) 
02 
O 
a/3/3 = -j (cosh a — cos /3) d—— sinh ad-sin /3 d— -|- cosh /3 r (Ax). 
op J 
ra" 
ca 
aa/3 = — (cosh a — cos /3) 
Oa op 
We may note that 
a (aa-/3/3) = (cosh a - cos /3) ( — - — + 1 ) (A x ), 
Of3 COL / 
0 2 02 
c c 
( 6 ) 
(7) 
* Love, ‘ Theory of Elasticity,’ p. 54. 
