Biomedical Engineering Reference
In-Depth Information
Z
X
Z
X
(a)
(b)
FIgure 24.2
(See color insert.)
Finite element models of the trabecular specimens with loading
and boundary conditions in compressive loading conditions in the longitudinal direction, respectively:
(a) trabecular specimen A and (b) trabecular specimen B.
Both geometrical nonlinearity and bone tissue material nonlinearity were considered in each
analysis. The bilinear tissue level constitutive model in Chapter 9 (Figure 9.1) was used to describe
bone tissue material nonlinearity. The four parameters in the model were
E
= 18 GPa,
E
u
= 5%
E
,
ε = 0.48%
c
T
(Bayraktar et al., 2004; Rho et al., 1997).
The cast iron plasticity material constitution in ABAQUS was used to describe the elastic-plastic
behavior of trabecular bone material with asymmetric yield strength and hardening in tension
and compression (Bayraktar et al., 2004; Gong et al., 2011). Linear analyses were done first to
determine the loading mode (tension or compression) of each element in each finite element mesh.
The maximum principal stress criterion was used to determine the loading mode of each element.
Suppose the three principal stresses of an element are
σ
1
,
σ
2
, and
σ
3
. If σσ
, and ε = 0.8%
t
T
>
3
, the element
1
<
1 3
, the element is predominantly compressive (Schileo et al.,
2008). In each nonlinear analysis, the initial apparent yield point was determined using the 0.2%
offset method.
is predominantly tensile; if σσ
24.3 reSultS from tHe model analySiS
Figure 24.3 shows the apparent stress-strain curves for the two trabecular cubes in compres-
sion and tension in the longitudinal and transverse directions. The linear portion of each curve
shows the elastic response and the slope gives the apparent Young's modulus in that direction. The
0.2% offset lines were drawn on the figure to determine the initial apparent yield point in each
loading case.
Table 24.2 lists the distribution of tissue von Mises stresses of the trabecular specimens and the
number of elements that yielded in compression and tension at the apparent yield point in each load-
ing direction. It was found that for both longitudinal and transverse loading, when loaded in tension,
very few elements yielded in compression and the majority of the yielding elements were in ten-
sion; but when loaded in compression, there were still a considerable amount of elements yielding
in tension. Figure 24.4 shows the von Mises stress distributions of the trabecular specimens at the
yield point in apparent compressive and tensile loading conditions in the longitudinal and transverse
directions, respectively.
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