Geoscience Reference
In-Depth Information
(see Chapter 9). Relevant parameters are measured in laboratory tests under
controlled conditions, and sometimes validated in the field. These parameters are
then used in numerical simulation models.
The triaxial stress and strain states are tensors. Their direction and size change
with orientation in such a manner that the internal equilibrium (of related forces)
and the internal compatibility (of related displacements) are satisfied. The triaxial
stress-strain relation is formulated as follows
xx
xy
xz
xx
xy
xz
=
S
or
ij = S ijkl
kl
(7.1)
yx
yy
yz
yx
yy
yz
zx
zy
zz
zx
zy
zz
where
ij and
ij are the stress and strain tensor in space.
ii represents a normal
ij is a shear stress (usually denoted by
).
ii represents a
stress and for i
j ,
normal strain and for i
ji . The
modulus S ijkl is a stiffness matrix, which compiles all possible interrelations, 81 in
total! Each component may be non-linear. Hence, drastic simplifications are
required to assess the triaxial mechanical behaviour in a practical way.
When considering a two-dimensional plane, the third dimension (perpendicular
to the considered plane) also plays a role. For example, if strain in the third
dimension is prevented (plane strain condition), then for an elastic material the
normal stress in the third dimension becomes, using (5.2):
j ,
ij is a shear strain, also denoted by
=
ij +
3 =
(
1 +
2 ), and the
isotropic stress becomes:
2 )/3. This may apply for a cross-sectional
two-dimensional view of elongate structures such as dikes and embankments.
Material properties for soil mechanics are usually obtained in a triaxial test, where
the stress symmetry is cylindrical. When interpreting test results sometimes a
correction is required to adjust for apparent field conditions.
The stress or strain state can be presented by a Mohr circle, which connects the
geometric orientation to the corresponding state. For a two-dimensional situation, it
is worked out in Fig 7.2. On the left-hand side (Fig 7.2a) the geometric situation of
an elementary soil volume is shown with actual stresses and displacements as they
work on the elementary volume. In the middle (Fig 7.2b) a particularly rotated
volume is shown subjected to the same state, where for deformations the distortion
and for stresses the shear are absent. The corresponding orientations are called
principal directions.
If the principal directions, here denoted by angle
i = (1+
)(
1 +
for
strain, are identical, the material is said to behave co-axially. It makes the
description of the behaviour, i.e. the relation between stresses and strains, easier
(the number of components in S ijkl decreases drastically). For sand, however, this is
less likely, because of density changes. Non-associativity can be expressed in the
so-called dilation angle, usually denoted by
for stress and angle
)
, which is related to the difference of
, see Chapter 9.
In Fig 7.2c, the state is shown in a compact way. Fulfilling internal compatibility
and equilibrium shows that the state for any orientation is a circle (after Mohr),
both for strain and stress. Every point on the circle reflects the state in a specific
orientation. The direction centre or pole of normals (DC 1 ) correlates this
and
)
 
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