CONTINUUM MECHANICS for ENGINEERS

Thus,
By eliminating ψ from this pair of equations we obtain
(6.8-20)
(6.8-21)
As already noted, the lateral surface of the shaft parallel to the x axis 3
must remain stress free, that is, the third of Eq 6.8-10 must be satisfied.
However, it is advantageous to write this condition in terms of the unit vector
ˆs along the boundary rather than unit normal as shown in Figure 6.8. It
follows directly from geometry that n1 = dx2/ds and n2 = –dx1/ds. In terms
of ds, the differential distance along the perimeter C, the stress components
in the normal direction will be given by
ˆn
which in terms of Φ becomes
( )
( 2 1)
∂Φ
∂Φ
= Gθψ, 1 −x2
; =− Gθψ, + x
∂x
∂x
2
2
(6.8-22)
(6.8-23)
Thus, Φ is a constant along the perimeter of the cross section and will be
assigned the value of zero here.
Finally, conditions on the end faces of the shaft must be satisfied. Beginning
with the first of Eq 6.8-12 we have in terms of Φ
∫∫
(6.8-24)
since Φ is constant on the perimeter. Likewise, by the same reasoning, the
second of Eq 6.8-12 is satisfied, while the third is satisfied since σ 33 = 0. The
first two conditions in Eq 6.8-15 are also satisfied identically and the third
becomes
1
Φ=−2Gθ dx2
dx
σ13 − σ23
=
ds ds
1 0
∂
+
∂
∂ Φ dx1
Φ dx dΦ
= =
x ds ∂x
ds ds
1
2
2 0
∂Φ
⎛ ∂Φ
⎞
dx1dx2 =
∂
⎜ dx
⎝ ∂
⎟ dx = Φ]
dx =
x ∫∫ x ⎠ ∫
2
2
2 1 1 0
⎛ ∂ ∂ ⎞
⎜−x
− x
⎝ ∂ ∂
⎟ =
∫∫ x x ⎠
dx dx M Φ Φ
1
2
1 2 t
1
2
b
a
(6.8-25)