Granular Materials in Civil Engineering: Recent Advances in the Physics of Their Mechanical Behavior and Applications to Engineering Works - Multiscale Geomechanics: From Soil to Engineering Projects - page 42

To be more precise in this comparison of internal actions, we have to be sure that
the internal movements that have been accounted for within the discontinuous
medium correspond to all the macroscopic internal movements observable when
considering the equivalent continuum. This requires the two following conditions to
be met:
− the macroscopic deformations in the continuum are due to intergranular sliding
within the discontinuous medium;

− the component of the macroscopic movements due to eventual rolling or
spinning movements at the grain scale can be neglected (no significant “roller
bearing” motion of the grains).

Kinematic analyses of numerical simulations by the discrete element method
[NOU 05] show that these assumptions are realistic as long as the grain shapes are
sufficiently irregular, which is the case for granular media used in civil engineering.
Therefore, we obtain complete equality between the mean values of the internal
actions within the discontinuous granular mass, and the corresponding equivalent
continuum:

1 1

π π π π

= = P dv (A)

[3.14]

V V

V

() A () A

As a result, the inner actions within the equivalent continuous medium follow the
same dissipation relation as the inner actions within the discontinuous medium.
Therefore, the phenomenological relationship found for the equivalent continuum
(equations [3.10] and [3.12]) can be seen as a direct explicit consequence at the
macroscale of energy dissipation due to friction within the discontinuous granular
mass.

3.1.3. Main practical consequences

3.1.3.1. Compatibility with heterogeneity
Under regular boundary conditions, when a granular medium evolution is
sufficiently close to the minimum dissipation, some specific properties are derived
from this proximity to a minimum. One of them is the compatibility of the internal
actions with the heterogeneity of the medium: dissipation relation [3.12] is verified
by the local variables as well as by the mean macroscopic values of the same
variables. With stress and strain rate heterogeneity, a different scale of local
fluctuations of the stresses and strain rates was shown [FRO 04], making the
dissipation equation [3.10] compatible with the macroscopic heterogeneity: this