PSI - Issue 61

Lívia Mendonça Nogueira et al. / Procedia Structural Integrity 61 (2024) 122–129 L. M. Nogueira et al. / Structural Integrity Procedia 00 (2024) 000–000

126

5

Through spectral decomposition, the fabric tensor M can be rewritten using its eigenvalues m i and corresponding eigenvectors m i in as follows:

3 i = 1

3 i = 1

m i ( m i ⊗ m i )

m i M i =

M =

(8)

This matrix representation has many remarkable properties which make this method especially attractive. Specifi cally, finding the major material orientation involves finding the principal axes given by the eigenvectors of the fabric tensor (Harrigan and Mann, 1984). In addition, the specimen symmetry can be related to the number of di ff erent calculated eigenvalues, i.e., orthotropic specimen to three distinct eigenvalues, transversely isotropic to two repeated eigenvalues, and isotropic when eigenvalues are all equal. Furthermore, the degree of structural anisotropy is also quantified by the ratio between the maximum and minimum eigenvalues (Cowin, 1986).

3. Results and discussions

The initial analysis aimed to assess the impact of pore shape and distribution within the RVE in microstructural analysis and quantification. The porous medium is comprised of two phases: the matrix (solid) and the pore (void). Four particular representative images generated such as to approximately resemble the real microstructures were taken into account:

• Case (a): randomly distributed circular pores; • Case (b): randomly distributed elliptical pores; • Case (c): oriented elliptical pores; • Case (d): highly oriented elliptical pores.

Each case was represented by synthetic binary images with 389 × 389 pixels resolution generated with a Python script to simulate RVE slices of an orthotropic material. The porosity was fixed in f p = 0 . 2 in all cases. The aspect ratio of the ellipses in Case (b) and Case (d) was set to R = 0 . 30, while for Case (c) it was set to R = 0 . 75. These aspect ratios take into account that the 2D images are representative of 3D models associated with oblate spheroid pores. Furthermore, for Case (b) and Case (c), the orientation of the ellipses was fixed at 50°. The convention used for the angle considers the reference with major semi-axis a aligned with the x-direction and minor semi-axis b aligned with the y-direction, with positive angle orientation in the clockwise direction. Using the MIL method implemented through BoneJ software for image processing and analysis (Doube et al., 2010), the heterogeneities orientation within the RVE is investigated. This analysis enables the identification of the representative ellipse to be further considered in the MT homogenization scheme for determining the e ff ective elastic properties. Figure 4 graphically presents the result from this analysis alongside the respective microstructure patterns used for each case. Table 1 presents the parameters of the ellipses obtained from the anisotropy analysis, where a represents the ellipse major semi-axis, b represents the ellipse minor semi-axis, R is the aspect ratio, i.e., the ratio b / a that defines the anisotropy degree, and θ denotes the orientation of the equivalent ellipse.

Table 1. Anisotropy analysis from BoneJ - f p = 0 . 2. R

a

b

θ (°)

Case (a) Case (b) Case (c) Case (d)

0.939 0.907 0.717 0.472

66.09 44.12 78.41 88.59

70.40 40.01 56.20 41.82

73.03

7.56

46.79 42.63

As inferred from these analyses, the representative ellipse of randomly distributed circular and elliptical pores (Case (a) and Case (b)) closely resembles a circle, with aspect ratio R substantially close to 1. This observation implies an equivalence to isotropic conditions. Therefore, for this observation scale of the RVE, it appears that pore shape does