32.8 The Application of Kröner’s Method to the Brazilian Test
433
or
N − iT = c
1 + 2 ln
R
R 2
+ ln
(σ − ζ 0 )
1
σ
− ζ 0
−
σ (1 − σ ζ 0 )
σ − ζ 0
,
(32.59)
where, as before, σ = e iϕ is an arbitrary point on the contour of the unit circle. By
transition in formula (32.59) to conjugate quantities, we obtain
N + iT = c
1 + 2 ln
R
R 2
+ ln
(σ − ζ 0 )
1
σ
− ζ 0
−
σ − ζ 0 )
σ (1 − σ ζ 0 )
.
(32.60)
We now solve the first main problem of the theory of elasticity for a circle
with contour loading (32.59). The functions of Muskhelishvili 3 (ζ ), , 3 (ζ )
for this task are already given in formulas (16.19) and (16.22). Substituting the
expressions (32.59)–(32.60) into these formulas, we find after calculating the
integrals
3 (ζ ) = c
1 −
ρ 2
0
2
+ ln
R
R 2
+ ln(1 − ζ 0 ζ ) −
1 − ρ 2
0
1 − ζ ζ 0
,
3 (ζ ) =
cζ
2
0 (2 − ρ 2
0 − ζ ζ 0 )
(1 − ζ ζ 0 ) 2
.
(32.61)
We obtain the Muskhelishvili functions (ζ ), ,(ζ ) for the circle |z| R, which
has a single wedge dislocation at the point with polar coordinates (ρ 0 , ϕ 0 ). To do
this, subtract the corresponding functions (32.61) from the functions (32.57). As a
result, we get
) = c
ρ 2
0 − 1
2
+ ln
ζ − ζ 0
1 − ζ ζ 0
+
1 − ρ 2
1 − ζ ¯
ζ 0
,
(ζ ) == cζ 0
1
ζ − ζ 0
+
ζ 0
1 − ζ ζ 0
+
(1 − ρ 2
0 )ζ 0
(1 − ζ ζ 0 ) 2
.
(32.62)
The functions (32.62) correspond to the stresses
1
c
τ rϕ =
2ρ 0 (1 − ρρ 0 )V 1 [ρ(1 + ρ 2
0 ) − ρ 0 V 2 ]
V 3 V 2
4
,
1
c
σ r = ρ
2
0 − 1 + ln
V 3
V 4
+
2(1 − ρ 2
0 )V 5
V 4
−
(1 − ρ 2
0 ) 2 (V 2
6 − ρ 2
0 V 2
1 )
V 7 V 2
4
,
(32.63)