Geoscience Reference
In-Depth Information
Let us consider two granular materials coming from the same homogeneous
mineral stock, compacted at the same density, with homothetic grain size
distributions, G 0 and G 1 , and characteristic diameters, D 0 and D 1 (for example D Max ).
The two materials are, then, geometrically similar in a ratio D 1 /D 0 :
− In order to mobilize the same internal friction within the two materials, the
maximum dilatancy rate has to be the same during shearing, so the amount of grain
breakage, or the probability of survival, also has to be the same, which means that
the stresses applied to the grains must verify the following relation of similitude:
{ σ grains (G 1 ) }= { σ grains (G 0 ) } x ( D 1 /D 0 ) - 3/m
[3.28]
The link between macroscopic stresses and stresses applied on the grains being
enforced by the geometrical similitude of the two materials, the macroscopic stress
states, necessary for mobilizing the same internal friction, must verify the second
relation of similitude, which is identical to the first one:
{ σ (G 1 ) }= { σ (G 0 ) } x ( D 1 /D 0 ) - 3/m
[3.29]
Equation [3.29] represents the scale effect rule which generalizes equation
[3.26]
2
.
In this expression, the scale effect rule resulting from grain breakage in Mode 1
is not fundamentally linked to a particular expression of the shear strength envelope.
Thus, it can be applied for different criteria, such as De Mello's or Hoek and
Brown's criteria.
3.2.2.3.2. Shear strength envelope τ = f n ,D): De Mello's criterion
If the expression of the shear strength envelope of material G 0 is given by
τ
= f
(,
σ
n D
)
[3.30]
ax
G
0
then, the shear strength envelope of material G 1 from the same mineral stock, having
a homothetic grain size distribution and the same density, is given by
3
3
[3.31]


D
m
D
m
=
1
σ
1
D


n
0
G
1

D

D
0
0
2 The result found here from a reasoning at the macroscale can also be demonstrated by a
reasoning at the microscale between two homothetic granular media by using the relations
that give the macroscopic stress function of the intergranular forces and the geometry of
granular arrangement (Weber's relationship).
 
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