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

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4

where f p represents the volume fraction of the pores (porosity), C m is the sti ff ness tensor of the matrix, I is the identity tensor, and D is the Hill tensor. To determine the sti ff ness tensor in the global coordinate system, the following equation is applied for basis transformation:

C θ MT = [ R ( θ,ϕ ) C MT R ( θ,ϕ ) T ]

(5)

where θ is a polar angle, ϕ is an azimuthal angle, and R ( θ,ϕ ) is the forth-order rotation matrix.

2.2. Microstructural analysis and quantification

The modeling of the microstructural architecture of a material with two distinct constituents, one dispersed in the other, has been accomplished using a second-rank fabric tensor. The concept lies in modeling through tensors both anisotropy and orientation of a material (Cowin, 2013). Among morphology-based methods, the Mean Intercept Length (MIL) was adopted thanks to the large amount of evidence supporting its appropriateness to predict mechanical properties in oriented materials (Moreno et al., 2014; Ketcham, 2005; Kahl et al., 2017). The concept is based on defining the average distance between the two phases or constituents, measured along a particular straight line. The traditional methodology for computing the MIL was first introduced by Whitehouse (1974) and Underwood (1973) based on the directed secants method summarized next. From planar binary images sections through a polished specimen of material, an array of parallel lines to a specified direction are traced, and the number of intersections between these lines and the interface between both phases is counted, as can be schematically seen in Fig. 3.

Fig. 3. MIL method procedures: (a) An array of parallel lines in θ direction is generated for a generated 2D binary image. In detail: the number of intercepts between the set of parallel lines and the interface between phases is counted; (b) Ellipse is fitted in the polar plot of the average value of the MIL for each direction. The Mean Intercept Length of a specific orientation, characterized by a unit vector n , is a scalar quantity calculated as follows:

l C ( n )

MIL ( n ) =

(6)

inwhich l is the total length of traced lines, and C ( n ) is the number of boundary hits in the direction specified by the vector n . In cases of partially oriented microstructures, Underwood (1973) and Whitehouse (1974) noted that by plotting MIL data on a polar graph and fitting it into an ellipse, the parameters of the ellipse could be linked to the material’s orientation, as shown in Fig. 3. From previous theoretical work given by Gibson (1985), and experimental work by Carter and Hayes (1977) and Bensusan et al. (1983), the fabric tensor M that describes the orientation and distribution of microstructure and the MIL tensor A was then related by Cowin (1986) as follows:

1 2

(7)

M = A −