CONTINUUM MECHANICS for ENGINEERS

where A 1 and A 2 are constants of integration depending on the boundary
conditions. Eq 6.9-3 written in terms of displacement derivatives has the
form
( )
σ = λδ u + µ u + u
ij ij kk , ij , ji ,
It follows by making use of Eqs 6.9-23 and 6.9-25 that
i j
σij λAδ ij µ A δij
A ⎡⎛
⎞ 3Axx⎤
2
2
= 3 1 + 2 ⎢⎜
1+
⎟ − 3
5
⎣⎝
r ⎠
⎥
r ⎦
(6.9-26)
(6.9-27)
Recall that the traction vector in the radial direction (see Eq 3.7-1) is
σ = σ nn which upon substitution of Eq 6.9-25 becomes
N ij i j
σ = 3λ + 2µ
A
N
( ) +
4µ
A
3
r
(6.9-28)
where the identity x has been used. Similarly, the tangential traction
i = rni
can be calculated using σ where the unit vectors νi are perpendic-
S = σijνν i j
ular to ni. The result is
σ = 3λ + 2µ
A
S
(6.9-29)
since ν . ix i = 0
The constants A1 and A2 are determined from boundary conditions on the
tractions. Clearly,
σ
σ
N
N
Carrying out the indicated algebra, we have the well-known formulas
σ N
σ S
(6.9-30a)
(6.9-30b)
These equations may be easily modified to cover the case where p 1 = 0, or
the case where p 2 = 0.
1
( ) +
1
2µ
A
3
r
=− p at r = r
1 1
=− p at r = r
3
pr 11 − pr
= 3 3
r − r
2
2 2
3
22
1
3
pr 11 − pr
= 3 3
r − r
2
3
22
1
3
rr 1 − 3
r
3
rr 1 +
2r
3
2
3
2
3
2
2
p − p
3
r − r
1 2
3
2 1
p − p
3
r − r
1 2
3
2 1