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 40

Figure 3.3. Theoretical minimum energy dissipation solutions

These results concern a region close to the theoretical minimum of the
dissipation, with a complete polarization of the relative movements at contact.
However, in the movements within usual granular media, a certain amount of
disorder is necessary to make kinematic compatibility possible. This leads us to
envisage larger domains of minimum dissipation having small but non-null values of
R(A) , as well as an incomplete material polarization. We have been able to analyze
heterogeneous movements in this enlarged framework [FRO 04].

Finally, let us denote “apparent friction”, ψ* , the scalar defined by:

sin ψ *

[3.8]

sin ψ =

1 − R . ( 1 s i n − ψ

() A

The dissipation equation [3.6] for the granular assembly becomes:

{ } { }

*

Tr P = sin ψ N P [3.9]

() A () A

This relation is formally identical to the one corresponding to the elementary
contact (equation [3.3]), in which the physical contact friction, ψ , is replaced by the
apparent friction, ψ* , which includes the population effect, R(A) .