Civil Engineering Reference
In-Depth Information
Q
Q
′
X
3
α
T(M)
P
P
′
O
X
2
X
1
Figure 1.4
Antiplane shear in an elastic medium. A vector
PQ
initially parallel to
x
3
becomes oblique with an angle
α
to the initial direction.
X
3
P
′
Q
′
P
Q
O
X
2
X
1
Figure 1.5
Longitudinal strain in the
x
3
direction.
vector
are
1
2
∂u
2
∂x
3
,
1
=−
2
=
3
=
0
(1.32)
The deformation is equivoluminal, the dilatation
θ
being equal to zero, and there is a
rotation around
x
1
.
Longitudinal strain
For this case only the component
e
33
of the strain tensor is different from zero. The
vectors
PQ
and
P
Q
are represented in Figure 1.5.
The stress tensor components that do not vanish are
σ
33
=
(λ
+
2
µ)e
33
(1.33)
σ
11
=
σ
22
=
λe
33
Unidirectional stress
From Equation (1.26) the stress component
σ
33
transforms a vector
PQ
parallel to the
axis
x
3
into a vector
P
Q
parallel to
x
3
. The ratio P
Q
/PQ is given by
P
Q
/
PQ
=
σ
33
/E
(1.34)
A vector
PQ
perpendicular to
x
3
is transformed in a vector
P
Q
parallel to
PQ
and
the ratio P
Q
/PQ is now given by
P
Q
/
PQ
=−
νσ
33
/E
(1.35)
