Civil Engineering Reference
In-Depth Information
p
X
3
p
p
V'
V
p
O
X
2
X
1
Figure 1.6
Compression of a volume
V
by a hydrostatic pressure.
Compression by a hydrostatic pressure
For this case, represented in Figure 1.6, the components of the stress tensor that do not
vanish are
σ
11
=
σ
22
=
σ
33
=−
p
(1.36)
From Equation (1.21) it follows that the dilatation
θ
is related to
p
by
p
λ
2
µ
3
θ
=−
+
(1.37)
−
p/θ
is the bulk modulus
K
of the material, which is equal to
The ratio
2
µ
3
K
=
λ
+
(1.38)
0and
θ
is nonzero. The deformation is
irrotational, as in the case with a longitudinal strain. Note that since a hydrostatic pressure
leads to a negative volume change, the bulk modulus
K
is positive for all materials and
in consequence Poisson's ratio is less than or equal to 0.5 for all materials.
Contrary to the case of simple shear,
=
1.6
Equations of motion
The total surface force
F
v
acting on the volume
V
represented in Figure 1.2 is
F
v
=
T
d
S
(1.39)
The projection of the force
F
v
on to the
x
i
axis is
F
v
i
=
(σ
1
i
n
1
+
σ
2
i
n
2
+
σ
3
i
n
3
)
d
S
(1.40)
S
