Page 211 - Advances in Biomechanics and Tissue Regeneration

206 10. DETERMINATION OF THE ANISOTROPIC MECHANICAL PROPERTIES OF BONE TISSUE

ρ app ¼ α w ρ cortical (10.2)
app
α b
Using the ρ app , it is possible to define the axial Young’s modulus, E axial . To define the transverse elastic modulus,
E trasnv , the relation between the ellipse minor axis length, β , the major axis length, β , and the axial elastic
min max
modulus, E axial , can be used, as Eq. (10.3) shows.

jjβ min jjE axial
jjβ max jj
E transv ¼ (10.3)
Poisson’s coefficient, ν, can be calculated according to the mixture theory using the relation between white and black
pixels, as represented in Eq. (10.4).

(10.4)
ðÞ
0:0 α b +0:3 α w Þ
ð
ν ¼
α t
with α t being the total number of pixels of the binary image I s . The shear modulus, G, can be expeditiously calculated
using Eq. (10.5).
E axial
(10.5)
G ¼
2ð1+ νÞ
Using the homogenized material properties (E axial , E trasnv , ν, and G), it is possible to define the constitutive matrix c ox y
0 0
for the ox y local coordinate system. Transforming c ox y with the transformation rotation matrix T, it is possible to
0 0
0 0
define the material constitutive matrix in the global axis.
10.3 VALIDATION

To validate this homogenization technique, some tests are performed, which allowed understanding the behavior of
the methodology used to acquire the fabric tensor. Thus, three numerical studies were tested, one concerning the influ-
ence of the size of the RVE, another related to scale analysis, and a third concerning the rotation effect of the RVE in the
acquisition of the θ ellipse parameter.

10.3.1 Scale Study

In order to understand the influence of the size of the RVE, three distinct models were constructed based on
Fig. 10.1A–C. The models presented in Fig. 10.8a and d are benchmark fabricated unitary binary images with a
well-defined material orientation, 0 and 45 degrees, respectively.
Alternatively, it was also used a realistic trabecular model RVE, represented in Fig. 10.8g and obtained from
Fig. 10.1C, a micro-CT binary image, was also used. The three models were repeated r n r n , being r n ¼ 1, 2, …, 10.
Applying the homogenization technique to all the RVE and corresponding repetitions and comparing the results
between each element of the constitute matrix (Fig. 10.9), it is perceptible that the scale of the RVE does not affect sig-
nificantly the acquisition of the mechanical properties. The small changes visible in Fig. 10.9C occur because the uni-
tary image, Fig. 10.8g, is not periodic, which means that the repetition of the image results in the creation of new
changes of phase.

10.3.2 Rotation Study

To understand the influence of the rotation in this methodology, the already presented RVEs were rotated with
respect to their initial position using an increment of 20 degrees in the interval between [0,180 degrees]. For the cases
presented in this chapter (Fig. 10.1), the average difference between the obtained material orientation and the expected
angle was of 5.83 degrees. This difference can be explained by the changes occurring on the source image upon the
rotation process, as can be observed in Fig. 10.10, where the red circle marks the changes in the same region in different
rotated images. In Fig. 10.11, it is perceptible that by applying a rotation to the image, the principal direction of the
fitted ellipse, θ, reflects the applied rotation.

I. BIOMECHANICS