Biomedical Engineering Reference
In-Depth Information
s
I
s
I
30
°
60
°
s
II
s
III
s
II
s
III
(a)
(b)
Figure 6.7
Schematic representation of periodic segments
in the octahedral plane: (a) 30
◦
symmetry segment; (b) 60
◦
symmetry segment.
s
I
s
I
Outer
bound
A
k
k
C
Inner bound
C
30
°
30
°
A
O
O
s
II
s
II
s
III
s
III
(a)
(b)
Figure 6.8
Schematic representation of outer (a) and inner
(b) bounds of a yield condition in the octahedral space.
Further restrictions for the shape of the yield condition in the octahedral plane
are based on the requirement of a convex shape
8)
, cf. Figure 6.8. The inner bound
corresponds to the Tresca yield criterion. An influence of the third invariant would
result in a deviation from the circular shape (e.g., von Mises).
Themaximumdistancebetweenthecircleandtheouterboundisobtainedfor
θ
=
30
◦
as (cf. Figure 6.8(b))
AC
k
=
AO
−
k
k
1
cos 30
◦
−
2
√
3
−
=
1
=
1
=
15
.
47%
(6.61)
of the circle diameter. The maximum distance between the circle and the inner
bound is obtained under the same angle as (cf. Fig. 6.8(b))
√
3
2
=
13
.
40%
AC
k
=
k
−
AO
k
=
1
−
cos 30
◦
=
1
−
(6.62)
8)
The convexity of the yield surface can be
derived from Drucker's stability postulate
[6, 24].
