AIRY STRESS FUNCTION FOR TWO DIMENSIONAL INCLUSION ...

u
in
θ
rS(
1 + κ 1)sin
2θ
= −
.
4(
κ µ + µ )
1 1
30
(3.3.30)
The above equations can be verified by substituting them into the equilibrium
equations i.e. (2.1.7).
Substitute the constants into equations (3.3.8) and (3.3.9) to obtain the final
stress and displacement equations for the part surrounding the circular inclusion i.e.
2
2
µ
κ µ
out S a S
a S 1
σ rr = − 2
+ 2
+
2 2r ( 2µ − µ + κ µ ) 2r ( 2µ
− µ + κ µ )
1 1
1 1
2
2
a Sµ
1
a Sκ
µ 1 1
2
− 2
+ S cos2θ
−
2r ( 2µ − µ + κ µ ) 2r ( 2µ
− µ + κ µ ) 2
1 1
1 1
4
2
4
2
3a Sµ
cos2θ
2a Sµ
cos2θ
3a Sµ
1 cos2θ
2a
Sµ
1 cos2θ
4
+ 2
+ 4
− 2
2r
( κ µ + µ ) r ( κ µ + µ ) 2r
( κ µ + µ ) r ( κ µ µ ) ,
+
1 1
1 1
1 1
2
2
2
µ
κ µ
out S a S a S
a S 1
σ θθ = + 2 − 2
− 2
2 2r 2r ( 2µ − µ + κ µ ) 2r ( 2µ
− µ + κ µ )
1 1
4
4
1 3a Sµ
cos2θ
3a Sµ
1 cos2θ
− S cos2θ
+ 4
− 4
2 2r
( κ µ + µ ) 2r
( κ µ + µ ) ,
1 1
1 1
4
2
out 1 3a Sµ
sin 2θ
a Sµ
2θ
τ rθ
= − S sin2θ
− 4
2
+
2 2r
( κ µ + µ ) r κ µ + µ
_
sin
( )
1 1
4
2
3a Sµ
1 sin 2θ
a Sµ
1 sin 2θ
4
− 2
2r
( κ µ + µ ) r ( κ µ + µ ) ,
1 1
1 1
1 1
1 1
1 1
2
2
2
a S rS rSκ a S
a S
out 1
µ
κ 1µ
ur
= − + −
−
+
4rµ 8µ 8µ 4rµ ( 2µ − µ + κ µ ) 4rµ ( 2µ
− µ + κ µ )
1 1
1
1 1 1
1 1 1
4
2
2
rS cos2θ a S cos2θ
a S cos2θ
a Sκ
1 cos2θ
− 3
+
+
+
4µ
4r
( κ µ + µ ) 4r(
κ µ + µ ) 4r(
κ µ + µ )
1
1 1 1
1 1
1 1
1 1
4
2
2
a Sµ
cos2θ
a Sµ
cos2θ
a Sκ
1µ
cos2θ
3
−
4r
µ ( κ µ + µ ) 4rµ
( κ µ µ ) 4rµ
( κ µ µ ) ,
−
+
+
1 1 1
1 1 1
(3.3.31)