170
6 Two-Dimensional Problems in Elasticity …
2 r dr = 2xdx + 2y dy
∴ dr =
x
r
dx +
y
r
dy
Also,
sec
2
θ dθ = −
y
x 2
x y +
dy
x
∂φ
∂ x
=
∂φ
∂r
∂r
∂ x
+
∂φ
∂θ
∂θ
∂ x
=
x
x 2 + y 2
∂φ
∂r
+
−
1
sec 2 θ
y
x 2
∂φ
∂θ
∴
∂φ
∂ x
= cos θ
∂φ
∂r
−
sin θ
r
∂φ
∂θ
Similarly,
∂φ
∂ y
=
∂φ
∂r
∂r
∂ y
+
∂φ
∂θ
∂θ
∂ y
∴
∂φ
∂ y
= sin θ
∂φ
∂r
+
cos θ
r
∂φ
∂θ
Now,
∂
2
φ
∂ x 2 =
cos θ
∂φ
∂r
−
sin θ
r
∂φ
∂θ
2
= cos
2
θ
∂
2
φ
∂r 2 −
2 sin θ cos θ
r
∂
2
φ
∂r ∂θ
+
sin
2
θ
r 2
∂
2
φ
∂θ 2
+
2 sin θ cos θ
r 2
∂φ
∂θ
+
sin
2
θ
r
∂φ
∂r
(i)
Similarly,
∂
2
φ
∂ y 2 = sin
2
θ
∂
2
φ
∂r 2 +
2 sin θ cos θ
r
∂
2
φ
∂r ∂θ
−
2 sin θ cos θ
r 2
∂φ
∂θ
+
cos
2
θ
r
∂φ
∂r
+
cos
2
θ
r 2
∂
2
φ
∂θ 2
(ii)
And,
∂
2
φ
∂ x∂ y
= −
sin θ cos θ
r
∂φ
∂r
+ sin θ cos θ
∂
2
φ
∂r 2 +
cos 2θ
r
∂
2
φ
∂r ∂θ