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sin
ψ
{}
{}
[3.6]
Tr
P
=
.
N
P
()
A
()
A
1
−
R
. ( 1 s i n
−
ψ
()
A
where function
R(A)
corresponds to this population effect and is related to an energy
exchange between neighboring moving contacts. We can call this function the
“internal feedback”. Comprised between 0 and 1, the internal feedback function is
related to the degree of disorder of the statistical distribution of the moving contact
orientations. Its mathematical representation verifies the following conditions:
N
p
()
A
aR
)
(1sin ) 1
−
ψ
=
−
()
A
{}
[3.7]
N
p
()
A
b
0
≤
R
≤
1
()
A
By comparing the dissipation rate,
Tr
{
P
(A
)}, with the input power, which is the
sum of the positive eigenvalues of tensor
P
(A
), we can show that the lower and
upper bounds of inequality [3.7b] define theoretical solutions to the dissipation
equation [3.6] in the cases of minimum and maximum relative energy dissipation.
For
R(A)=
0, i.e. for solutions corresponding to minimal dissipation, the granular
assembly follows the same dissipation equation as the elementary contact (see
equation [3.3]).
Minimum dissipation solutions correspond to ordered patterns in the distribution
of moving contact orientations within the granular assembly, consisting in two 3D
modes with signatures (+,-,-) and (+,+,-), separated by a plane strain border mode
(+,0,-). In plane strain condition, this pattern coincides with Rankine's slip lines
(1857). These theoretical solutions imply a complete polarization of the sliding
contact orientation distribution at a given time.
These theoretical results, combined with various experimental observations,
suggest that the discontinuous granular assembly verifies a “minimum dissipation
rule”, which can be expressed as follows:
under regular, monotonic, quasi-
equilibrium boundary conditions, the moving medium tends toward a regime of
minimum energy dissipation compatible with the imposed boundary conditions; this
regime is independent of the initial particular conditions
. For granular media in
slow motion, such a rule has been shown [FRO 04] to be a corollary of the minimum
entropy production theorem based on the thermodynamics of dissipative systems
near equilibrium [PRI 77].
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