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equation obtained for the local variables also holds for the mean values.
Furthermore, it has been demonstrated that heterogeneities in internal movements
and internal forces compatible with the different forms of the dissipation equation
represent the eventual patterns of strain localization or stress concentration
[FRO04].
This compatibility of the energy dissipation relation with the heterogeneity
justifiesconsiderationofthemeanvaluesofthestressandstrainrates.
3.1.3.2. Dilatancyrule:theeffectsoftwo-waycyclicloading
i α= && for i 0
&
Introducing the auxiliary variables
εε
ε ≠
, equation [3.10]
becomes:
&
1s in* 0
.σ ε =
[3.15]
i αψ
ii
i
Under axisymetric loading, assuming that the radial strains have the same sign
(so 2
α= ),weobtain:
3
σ
v 1s in*
1s in*
αψ
αψ
[3.16]
1
=−
1
3
σ
3
1
1
whereindex1correspondstotheaxialdirection.
For the usual triaxial compression test, 1
α =+ and 3
1
α =− ;equation [3.16]
1
becomes:
σ
πψ
*
42
v
2
[3.17]
1
=−
1
. ta n
+
σ
3
1
Equation [3.17], directly derived from the dissipation equation [3.10], appears to
be Rowe's dilatancy rule in axisymetric compression. We can see that equation
[3.17] gives the expression of the mobilized shear as a product of two factors: the
dilatancyrateandtheapparentintergranularfrictioncoefficient.
If, at a given point of the loading in compression, we reverse the strain direction
in order to impose loading in extension, the change in the sign of i
ε & in
equation[3.10]also producesasign-changein theauxiliaryvariables i α in equation
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