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Under axisymmetrical stress or plane strain conditions, the dilatancy equation
established above leads to the following expression of the failure criterion:
Sup
Inf
(, ,)
tan (
σσσ
=
π
*
42
ψ
2
1
2
3
+
)
[3.18]
(, ,)
σσσ
1
2
3
For more general 3D conditions, we can demonstrate that at critical state:
− equation [3.18] gives the minimum in strength and in dissipation under usual
boundary conditions at the same time;
− the corresponding strain mode is the plane strain mode in the direction of the
intermediate principal stress.
These conditions define the well-known Coulomb's criterion with precision.
3.1.3.6.
Link with fluid mechanics
Another interesting result from the use of this approach to energy dissipation by
friction can be stressed: if the friction between grains tends toward zero, the
dissipation vanishes within the medium and, as a consequence, the macroscopic
apparent friction also vanishes. The expression of the failure criterion shows that,
under this condition, the stress states converge toward isotropy, even at large
deformations; the dissipation relation means that, in the vicinity of an isotropic state
of stress, the specific volume becomes constant. A remarkable solution therefore
arises for this particular behavior, which includes no dissipation during the
movement, isotropic states of stress and a constant specific volume for any given
state of deformation: this is the behavior of a “perfectly incompressible fluid”,
which is the basis of hydraulics. This result appears to be quite understandable,
because the granular medium in motion is in this case an assembly of grains sliding
without friction against each other while staying in contact: this is the classical
microscopic description of a liquid.
3.1.3.7.
Strain localization - shear band internal structure
A strong compatibility with the heterogeneity being inscribed in the energy
dissipation relation, the strain localization in shear bands can be studied using the
internal actions without particular caution. The macroscopic shear bands found with
this approach tend to converge toward a plane strain kinematic, with an angle close
to
(
π
−
with the direction of the major principal stress [FRO 04] (see
Figure 3.6).
4
2
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