Geology Reference
In-Depth Information
In spite of the applied stress being uniaxial, the material is also strained in the
two directions orthogonal to the stress axis. The ratio of the magnitudes of these
strains to the axial strain is given by the dimensionless
Poisson's ratio
,
λ
2 (λ
+
μ)
.
σ
=
(1.255)
If the state of stress is hydrostatic, there are no shear stresses, since by definition
a fluid in equilibrium cannot support shear stress, and τ
ij
= −
i
j
, where
p
is the
hydrostatic pressure. Then, from the relation (1.251) between strain and stress,
p
δ
λ
2μ(3λ
+
p
2μ
δ
p
3λ
+
i
j
i
j
i
e
ij
=−
2μ)
(
−
3
p
)δ
−
=−
2μ
δ
j
.
(1.256)
The dilatation is
2μ). The ratio of the pressure to the negative
of the dilatation measures the resistance of the fluid to compression and is called
the
bulk modulus
, or, as is common in geophysical usage, the
incompressibility
,
K
.
It is
Θ=
e
kk
=−
3
p
/(3λ
+
p
−Θ
=
2
3
μ.
=
λ
+
K
(1.257)
The dilatation and the strains are, in turn, connected to the displacement field by
∂
u
j
∂
x
i
+
1
2
∂
u
i
∂
x
j
Θ=
e
kk
=∇·
u
and
e
ij
=
.
(1.258)
In terms of the displacement field, the expression for the stress, given by Hooke's
law (1.245), becomes
∂
u
j
∂
x
i
+
∂
u
i
∂
x
j
i
j
τ
ij
=
λ
∇·
u
δ
+
μ
.
(1.259)
Betti showed in 1872 (see Love (1927), pp. 173-174) that for two systems of
surface tractions and body forces, the work done by the first system, acting through
the displacements caused by the second system, is equal to the work done by the
second system, acting through the displacements caused by the first system. Let the
surface tractions per unit area be
t
i
and
t
i
, the body forces per unit volume be
F
i
and
F
i
and the displacement fields they cause be
u
i
and
u
i
, respectively. Then the
work done by the first system of surface tractions and body forces, acting through
the displacements caused by the second system is
t
i
u
i
dS
F
i
u
i
dV
S
τ
ji
ν
j
u
i
dS
F
i
u
i
dV
,
+
=
+
(1.260)
S
V
V
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