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 44

[3.16]. We therefore obtain a stress−strain relationship similar to equation [3.17],

( ) ( )

πψ
-

* πψ

+ . If the

*

but with a different coefficient:

2

instead of

2

tan tan

42 42

direction of loading is changed again, returning to compression, equation [3.17] will
hold again. Therefore, the energy dissipation equation [3.10] predicts that, under
two-way loading, the stress−strain relationship depends on the dilatancy rule, which
oscillates between two conjugate relations dependant on the sign of the loading
direction. This behavior, which is well established experimentally, is the
consequence of the irreversibility of the deformation process resulting from energy
dissipation by friction. The response during unloading is not the exact inverse of the
response during loading, and the difference between the two responses is due to the
energy dissipation by friction.

An important practical consequence is the fact that, during a two-way loading
close to the isotropic stress state, the volume change due to a cycle is always
contractive and the cyclic loading creates a densification of the granular medium
[FRO 83]. This is the basic principle of compaction by two-way cyclic loading.

Finally, under plane strain condition, for the same reason, equation [3.10] leads
to similar forms of the dilatancy rule [FRO 01]. Under two-way cyclic loading, the
alternative response of the medium between two different dilatancy rules in loading
and unloading has been demonstrated by numerical discrete simulations [NOU 05]
for grains with sufficiently irregular shapes.

3.1.3.3. Experimental validation from triaxial test results
Figure 3.4a shows an experimental validation of the dilatancy rule
(equation [3.17]) for a triaxial compression test on crushed limestone with very
angular particles of irregular shape at three different initial densities and under the
same confining pressure. The stress−strain curves in the lower left-hand diagram and
the volume change in the upper left-hand diagram show the influence of the initial
density on the material behavior, especially at peak strength. The dilatancy diagram
on the right-hand side shows a linear relation between the stress ratio, σ 1 /σ 3 , and the
dilatancy rate, which is in agreement with equation [3.17]. The slope of this straight
line gives a value of the apparent friction ψ* = 42.5°. This straight line is
independent of the initial density, in agreement with the least dissipation rule.

We observe that the three samples exhibit the same contractancy rate at the
beginning of the loading. This is also in agreement with equation [3.17] in the
vicinity of an isotropic stress state.