Geoscience Reference
In-Depth Information
The fractional increase in volume caused by a deformation is called
cubical dilatation
and
is written
. The volume of the original rectangular parallelepiped is
V
,where
V
=
x
y
z
(A2.14)
The volume of the deformed parallelepiped,
V
+
V
,isapproximately
V
+
V
=
(1
+
e
xx
)
x
(1
+
e
yy
)
y
(1
+
e
zz
)
z
(A2.15)
The cubical dilatation
is then given by
change in volume
originalvolume
=
V
+
V
−
V
=
V
(1
+
e
xx
)(1
+
e
yy
)(1
+
e
zz
)
x
y
z
−
x
y
z
(A2.16)
=
x
y
z
Therefore, to first order (recall that assumption of infinitesimal strain means that products of
strains can be neglected), the cubical dilatation is given by
=
e
xx
+
e
yy
+
e
zz
or
∂
u
x
+
∂v
y
+
∂
w
∂
=
∂
∂
z
or
=
∇·
u
(A2.17)
The relationship between stress and strain
In practice, in a given situation, we want to calculate the strains when the stress is known. In
1676, the English physicist Robert Hooke proposed that, for small strains, any strain is
proportional to the stress that produces it. This is known as
Hooke's law
and forms the basis of
the theory of perfect elasticity. In one dimension
x
, Hooke's law means that
σ
ce
xx
where
c
is a constant. Extending the theory to three dimensions gives thirty-six different
constants:
=
xx
σ
xx
=
c
1
e
xx
+
c
2
e
xy
+
c
3
e
xz
+
c
4
e
yy
+
c
5
e
yz
+
c
6
e
zz
(A2.18)
...
...
c
36
e
zz
If we assume that we are considering only isotropic materials (materials with no directional
variation), the number of constants is reduced from thirty-six to two:
σ
=
c
31
e
xx
+
c
32
e
xy
+
c
33
e
xz
+
c
34
e
yy
+
c
35
e
yz
+
zz
σ
=
(
λ
+
2
µ
)
e
xx
+
λ
e
yy
+
λ
e
zz
xx
=
λ
+
2
µ
e
xx
σ
yy
=
λ
+
2
µ
e
yy
σ
=
λ
+
2
µ
e
zz
zz
σ
xy
=
σ
yx
=
2
µ
e
xy
(A2.19)
σ
xz
=
σ
zx
=
2
µ
e
xz
σ
=
σ
=
2
µ
e
yz
yz
zy
