Geoscience Reference
In-Depth Information
xx,t
= u
x,xt
;
xy,t
= u
x,yt
;
yx,t
= u
y,xt
;
yy,t
= u
y,yt
(9.26)
Volumetric strain rate is described by
,t
=
xx,t
+
yy,t
, distortion strain rate by
,t
=
u
x,yt
u
y,xt
and the (free body) rotation rate by
2
,t
=
½(
u
x,yt
u
y,xt
). For rigid plastic
flow the condition of constant volume is adopted,
yy
=
0, at any time. The
xx
deviator strain rate
D
,t
is defined by
yy,t
)
2
2
2
D
,t
=
[(
] > 0
(9.27)
xx,t
,t
of the principal direction, the strain rate components
can be expressed in terms of
D,
Together with the angle
)
)
and
2
according to
xx,t
=
D
,t
cos(2
)
) and
yy,t
= D
,t
cos(2
)
) ;
(9.28a)
xy,t
= D
,t
sin(2
)
)
2
and
yx,t
= D
,t
sin(2
)
)+
2
(9.28b)
For a material with a von Mises yield criterion, with maximum deviator
k
, where
k
is a
constant cohesion (no material friction), the related stress state is expressed in
terms of the principal direction
,
k
and the isotropic stress
p
, according to
xx
=
p
k
cos(2
)
) and
yy
=
p
k
cos(
)
)
(9.29a)
xy
= k
sin(2
)
) and
yx
= k
sin(2
)
)
(9.29b)
The work rate by the plastic flow is defined by “force times corresponding
distance covered in a time step”. And so, for all components in all elements of
volume
V
, the work rate is
W
,t
=
ij
ij,t
dV
. Elaboration yields
W
,t
=
(
p
k cos
(2
)
))(
D
,t
cos(2
)
))
(
p
k
cos(2
)
))(
D
,t
cos(2
)
))
(
k
sin(2
)
))(
D
,t
sin(2
)
)
2
)
(
k
sin(2
)
))(
D
,t
sin(2
)
)
2
)
dV
D
,t
(2cos
2
(2
2sin
2
(2
= k
)
)
)
))
dV
yy,t
)
2
,t
2
]
dV
= 2k
D
,t
dV = k
[(
(9.30)
xx,t
At the border (interface
A
) of the plastic domain, where the adjacent boundary is
rigid and the yield stress
k
and a jump velocity
u
,t
act, the local work rate is
W
,t
= k
|
u
,t
| dA
(9.31)
W
,t
. This will
be used in determining visco-plastic elements. Notice, that there is no effect of
isotropic stress
p
and since
The total internal plastic work rate is therefore expressed by
W
,t
xy
=
yx
also no effect of body rotation
2
.
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