Geoscience Reference
In-Depth Information
7.2.1.2.
Plastic part
p
d
Δ
are defined as
The shear force
T
and the rate of plastic sliding
) (
2
2
2
2
p
p
p
Tf
=
+
d
Δ=
d
+
d
and
[7.5]
The yield function is assumed to be of Mohr-Coulomb type:
()
p
Ff
(,)
κ
=− Δ=
T f
κ
0
[7.6]
n
P
κΔ
i s ni otropic hardening/softening parameter. The hardening
function is defined by a hyperbolic curve in the
()
where
p
κ −Δ
plane, which involves two
material constants:
φ
and
k
.
p
0
tan
tan
φ
Δ
p
p
0
p
κ
=
[7.7]
p
φ
+Δ
n
p
p
0
The value of
κ
asymptotically approaches
tan
φ
. The initial slope of the
k
hyperbolic curve is
.
p
0
The elastic sliding behavior between two particles does not have a coupling
effect (i.e. there are no shear-induced normal movements). However, plastic sliding
often occurs along the tangential direction of the contact plane with an upward or
downward movement; thus, shear dilation/contraction work, due to plastic
movements, to the friction loss of contact planes. The relationship between normal
and tangential displacements is governed by the dilatancy equation:
p
n
p
d
T
=−
Δ
tan
φ
[7.8]
0
d
n
where φ
0
is a material constant which, in most cases, can be considered equal to the
internal friction angle φ
μ
. This equation can be derived by equating the dissipation
work due to plastic movements to the friction loss of contact planes in the same
orientation.
On the yield surface, under a loading condition, the shear plastic flow is
determined by a normality rule applied to the yield function. However, the plastic
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