CONTINUUM MECHANICS for ENGINEERS

Since σ 33 = 0, the third integral here is trivial. Considering the first integral
we may write
⎧ ∂
∂
⎫
Gθ ( ψ, 1−x2) dx1dx2 = Gθ
x1( ψ, 1−x2) + x ψ,
+ x dx dx
∫∫ ∫∫ ⎨
( ) ⎬
⎩∂x1
∂x2
⎭
(6.8-13)
2
where the condition ∇ ψ = 0 has been used. Green’s theorem allows us to
convert to the line integral taken around the perimeter C
∫c
(6.8-14)
which by Eq 6.8-11b is clearly satisfied. By an analogous calculation we find
that the second integral of Eq 6.8-12 is also satisfied.
On the end faces of the shaft, x3 = 0 or x3 L , the following conditions
must be satisfied
=
Again, since σ 33 = 0, the first two of these are trivial. The third leads to
Defining the torsional rigidity as
[ ]
Gθ x ψ − x n ψ x n ds
∫∫ ∫∫ ∫∫
[ ]
{ ( ) + ( + ) } =
1 , 1 2 1 , 2 1 2 0
1 2 1 1 2
( ) =
x2σ33dx1dx2 = x1σ33dx1dx2 = 0;
x1σ23 −x2σ13
dx1dx2 Mt 2 2
M = Gθ x + x + xψ −x
ψ dx dx
t
∫∫
( 1 2 , , )
1 2 2 1 1 2
( 2 , , )
∫∫ 1
2 2
K = G x + x + x ψ −xψ
dx dx
( )
1 2 2 1 1 2
(6.8-15)
(6.8-16)
(6.8-17)
which can be evaluated once ψ x1, x2
is known, we express the angle of
twist as
Mt
θ =
K
(6.8-18)
A second approach to the general torsion problem rests upon the introduction
of a torsion stress function, designated here by Φ and defined so
that the non-zero stresses are related to it by the definitions
σ13 = σ23
∂
=−
∂
∂
Φ Φ
;
x ∂x
2
1
(6.8-19)