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flow in the direction perpendicular to the contact plane is governed by the
stress−dilatancy equation in equation [7.8]. Thus, the flow rule is non-associated.
7.2.1.3. Interlocking influence
The internal friction angle φ μ is a constant for the material. However, the peak
friction angle, φ , on a contact plane is dependent on the degree of interlocking by
neighboring particles, which can be related to the state of the packing void ratio e
by:
m
= 
e
e
tan
φ
c
tan
φ
[7.9]
p
μ
where m is a material constant [BIA 94].
φ is greater than φ . When the
packing structure dilates, the degree of interlocking and the peak frictional angle are
reduced, which results in a strain-softening phenomenon.
For dense packing, the peak frictional angle p
The dilative behavior of a granular material does not depend on the absolute
value of the assembly void ratio, but rather on its relative value compared to the
critical void ratio. Under critical state, the granular material will remain at a constant
volume, while it is subjected to a continuous distortion. The void ratio
corresponding to this state is e .
The critical void ratio e c is a function of the mean stress. The relationship has
traditionally been written as:
=−
p
()
=Γ−
λ
log
p
or
ee
λ
log
[7.10]
c
r e f
p
ref
where Γ and λ are two material constants and p' is the mean stress of the packing,
and ( ref
e
p
, ref
) is a reference point on the critical state line.
7.2.1.4. Local elasto-plastic relationship
With the elements discussed above, the final incremental stress−strain relation of
the material can be derived that includes both elastic and plastic behavior, given by
 
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