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Figure 3.2.
The concept of Internal Actions
3.1.2.
Fundamentals
3.1.2.1.
Elementary intergranular contact
Let us consider a simple contact,
c
, between two grains,
a
and
b
, sliding with a
relative velocity,
(/)
a
f
, with an elementary friction
angle at contact,
ψ
(see Figure 3.2a). The elementary law of friction gives the
following relation between these two vectors, which corresponds to the energy
dissipation during sliding:
v
, under a contact force,
(/)
ab
.
=
sin .
ψ
[3.1]
(/) (/)
ab
ab
(/) (/)
ab
ab
From the product of these two vectors, we can define a “tensor of elementary
contact action”,
p
(c), which is a symmetrical second-order tensor, its trace being the
mechanical work rate produced by the contact force
(/)
f
during sliding:
ab
[
]
p=
1
2
( )
(/) (/) (/)
a b
⊗
a b
+
a b
⊗
(/)
a b
p
=
f v
+
v f
orincomponents
1
2
[3.2]
ij
i
j
i
j
.
{}
Tr
p
=
then
()
( / )
a b
( / )
a b
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