Biomedical Engineering Reference
In-Depth Information
s
I
s
I
y
P(
s
I,
s
II,
s
III
)
P
Hydrostatic
axis
N (
s
m
,
s
m
,
s
m
)
q
J
′
3
3
3
cos 3
q =
1
J
1
2
′
2
)
3/2
(J
2
3
3
s
III
O
N
x
Octahedral plane
s
II
s
III
s
II
(a)
(b)
Figure 6.6
Geometrical interpretation of basic stress invari-
ants in relation to an arbitrary stress state
P
: (a) principal
stress space; (b) octahedral plane,
J
1
= const. (view along
the hydrostatic axis).
an octahedral plane. A dependency of the yield condition on
J
3
results in a deviation
fromthecircleshape.Adependencyon
J
1
denotes a size change of the cross
section of the yield surface along the hydrostatic axis. However, the shape of the
cross section remains similar in the mathematical sense. Therefore, a dependency
on
J
1
can be represented by sectional views through planes along the hydrostatic
axis. The geometrical interpretation of stress invariants [13] is given in Figure 6.6.
It can be seen that an arbitrary stress state
P
can be expressed by its position
along the hydrostatic axis (
√
3
J
1
)
=
ξ
and its polar coordinates (
2
J
2
=
ρ
,
θ
(
J
2
,
J
3
))
in the octahedral plane through
P
. For the set of polar coordinates, the so-called
stress Lode angle
1
60
◦
as [23]
θ
is defined in the range 0
≤
θ
≤
3
√
3
2
J
3
(
J
2
)
3
/
2
cos(3
θ
)
=
·
(6.58)
The set of coordinates (
ξ
,
ρ
,cos(3
θ
)) is known as the Haigh-Westergaard coordi-
nates.
To investigate the shape of the yield surface, multiaxial stress states must be
simulated and the shape within the corresponding octahedral plane has to be drawn
according to the following transformation, which projects a principal stress state
first in the octahedral plane (angle of transformation: cos
ϑ
=
1
/
√
3) and then to
the Cartesian coordinate system (
x
,
y
) shown in Figure 6.6(b).
√
6
·
σ
I
−
0
.
5
·
(
σ
II
+
σ
III
)
2
y
=
(6.59)
1
√
2
·
(σ
III
−
σ
II
)
x
=
(6.60)
In such a case, it is essential whether the plastic behavior is pressure sensitive,
that is, depends on
J
1
, or not. If there is no dependency on
J
1
, that is, a constant
