Biomedical Engineering Reference
In-Depth Information
The flow rule is in general stated in terms of a function
Q
,whichisdescribed
in units of stress, and is called a
plastic potential
.Withd
λ
, a scalar called
plastic
multiplier
, the plastic strain increments are given by
λ
·
∂
Q
∂
σ
d
ε =
d
(6.63)
The flow rule given in Eq. (6.63) is said to be associated if
Q
F
, otherwise it is
nonassociated. Hardening can be modeled as isotropic (i.e., initial yield surface
expands uniformly without distortion and translation as plastic flow occurs) or
as kinematic (i.e., initial yield surface translates as a rigid body in stress space,
maintaining its size, shape, and orientation), either separately or in combination.
With the isotropic hardening parameter
κ
and the kinematic hardening parameters
α ={
α
=
T
can be composed and the
yield condition under consideration of hardening effects results as
x
,
...
,
α
}
T
, the hardening vector
q
={
κ
,
α}
xz
F
=
F
(
σ
,
q
)
=
0
(6.64)
Equation (6.64) can be further extended, for example, by incorporating damage
effects or anisotropy (fabric-dependent criterion), [25, 26]. More details onmodeling
of plastic material behavior can be found in the standard topics [22, 27-29].
6.4
The Structure of Trabecular Bone and Modeling Approaches
At the first glance, the bone may look as a solid dense material. However, most
of the bones rather form some kind of composite material where the outer shell
is made up of a dense layer, the so-called
compact bone
9)
and a core made up of a
cellular structure, the so-called
trabecular bone
10)
, cf. Figure 6.11. These two types of
bones are distinguished based on their porosity
11)
and microstructure. Trabecular
bone is much more porous with porosity ranging anywhere from approximately 50
to 90% whereas the cortical bone is much denser with a porosity ranging between
5 and 10%. According to Gibson and Ashby [30], any bone with a porosity larger
than 30% is classified as ''cancelous.'' A typical range for the density of trabecular
bone is between 94 kgm
−
3
(extremely low density) and 780 kgm
−
3
(moderately
1900 kgm
−
3
, [11, 31].
high), [11]. Compact bone has a density of about 1800
...
9)
Also known as cortical bone; Latin name:
substantia corticalis.
10)
Also known as cancellous or spongy bone;
Latin name: substantia spongiosa or sub-
stantia spongiosa ossium.
11)
The porosity is defined as the fraction of
the volume of void-space and the total or
bulk volume of material, including the solid
and void components. The range is be-
tween 0 and 1, or as a percentage between 0
and 100%.
