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model with five intrinsic parameters. The stress-strain behaves linearly in the
elastic range, involving Young's modulus E and Poisson's ratio
. Two parameters
define the failure criterion (flow potential F ): friction angle
and cohesion c , by F
= |
|
c
' tan
, and G = |
|
c
' tan
is the plastic potential. The dilation
angle
applies to cover irreversible changes in volume due to shearing. In the case
of associated behaviour
(Drucker's postulate), it is found for frictional
materials that plastic deformations evolve with an increase of volume (dilatancy),
which is not realistic. In the standard Mohr-Coulomb model the non-linear
elasticity of soils is covered by an average linear elasticity E 50 , representing the
stress-strain rate after 50% of the failure stress. The dilation angle
=
, which
describes the volume strain at failure, contributes substantially to the strength,
particularly in sands. The actual friction angle and the dilation angle are
interrelated, and
cv is the friction angle at
constant shear-induced volume change (critical state, critical density), which is
roughly equivalent to the angle of repose of a loose dry material (ranges from 30º
to 37º), see Fig 3.4b. It depends on mineralogy, grading and shape, but not on the
test type. A common value for sands is
=
cv + 0.82
is used (Bolton). Here,
cv = 33º. In the perfectly-plastic approach
the dilation angle
is usually kept constant. In reality, the value changes with the
shear strain level, i.e. finally it tends to zero (Fig 9.1b).
The modified Mohr-Coulomb model adopts a non-linear shear modulus,
according to
G = G 0 ( p'/p' 0 ) 1-)
(9.6)
is
usually around 0.4. For the plastic strain dependency of the actual dilation angle
Here, G 0 is a reference value at isotropic effective stress p' 0 . The value of
)
and the actual friction angle
in the modified Mohr-Coulomb model, the
following formulas are adopted
sin
= sin
0 exp(
pl /
;
)
(9.7a)
sin
= (sin
cv + sin
)/(1 + sin
sin
cv )
(9.7b)
Here,
0 is the initial value,
cv the value at constant shear-induced volume
change,
pl the equivalent plastic strain (changing during the numerical calculation)
and
is in the
order of 0.07). Adopting a kinematic yield cap in the compression domain is
another adjustment (see literature).
For coarse soils, Steenfelt emphasised the importance of taking the strength and
dilation parameters not as an easy matter. He showed practical examples and
suggested that Bolton's simplified expressions are appropriate.
;
a parameter to be determined by long-duration triaxial testing (
;
o
plane:
0
.
5
I
(9.8a)
pl
cv
max
R
o
triaxial:
0
.
3
I
(9.8b)
tr
cv
max
R
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