Biomedical Engineering Reference
In-Depth Information
Table 6.7
Generalized linear elastic Hooke's law and
independent material constants.
Type
Number constants
Anisotropic
21
Orthotropic
9
Transverse isotropic
5
Orthotropic, cubic structure
3
Isotropic
2
flow occurs. The hardening rule describes how the yield criterion is modified
by straining beyond the initial yield. In the following section, the mathematical
and graphical representations of the initial yield criterion are discussed in detail.
The initial yield criterion can generally be expressed for isotropic and anisotropic
materials as
F
=
F
(
σ
ij
)
(6.56)
o
ij
and deviatoric
part
σ
ij
. For an isotropic material, the stress state can then be expressed in terms
of combinations of three independent stress invariants. In the following, the set of
the so-called
basic
invariants,
J
1
,
J
2
,and
J
3
,isused,where
J
1
is the first invariant of
the spherical stress tensor (
wherethestateofstress
σ
ij
can be split into its spherical part
σ
ij
)and
J
2
and
J
3
are the second and third invariants
o
σ
σ
ij
), [20, 21]. Further sets of independent invariants can be
found, for example, in Altenbach
et al
. [22].
Thus, one can replace Eq. (6.56) for an isotropic material by
of the deviatoric tensor (
F
(
J
1
,
J
2
,
J
3
)
F
=
(6.57)
The yield condition
F
0 represents a hypersurface in the
n
-dimensional stress
space (in the case of isotropic materials,
n
is equal to the six independent
stress tensor components) and is also called the
yield surface
.Adirect
graphical
representation of this yield surface in a Cartesian coordinate system with three
coordinates is not possible due to its dimension. However, a reduction of the
dimensions is possible if a principal axis transformation is applied to the argument
σ
=
ij
. The components of the stress tensor are then uniquely reduced for isotropic
materials to the principal stresses
III
on the principal diagonal of the
stress tensor. In such a principal stress space, it is now possible to graphically
represent the yield condition as a three-dimensional surface. A hydrostatic stress
state (
σ
I
=
σ
II
=
σ
III
) lies in such a principal stress system on the space diagonal
(hydrostatic axis). Any plane perpendicular to the hydrostatic axis is called an
octahedral plane
. The particular octahedral plane passing through the origin is
the deviatoric plane or
π
-plane [20]. On the basis of the dependency of the yield
condition on the invariants, a descriptive classification can be performed. Yield
conditions independent of hydrostatic stress can be represented by the invariants
J
2
and
J
3
. Stress states with
J
2
= const. lie on a circle around the hydrostatic axis in
σ
I
,
σ
II
,and
σ
