PSI - Issue 80

Pavel Šandera et al. / Procedia Structural Integrity 80 (2026) 169–176 Šandera / Structural Integrity Procedia 00 (2025) 000 – 000

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Appendix A. The model of crack in porous metals A.1. Creating variants of the regular pore structure

The primary lattice in which the pores are arranged is oriented so that the edges of the cubes lie in the coordinate axes. For individual calculations, many variants are created using translational and rotational transformations of the lattice. For a plastic zone smaller than 50 μm, t he transformation system contains 4 different displacements in the x -axis direction and 3 in the y -axis direction and, for larger zones, 5 different displacements in the x -axis direction and 3 in the y -axis direction. This is followed by a rotation of the whole grid around the y -axis by an angle α (the original position plus 6 non-zero values) and then around the x -axis by an angle β (the original position plus 4 non-zero values), which represents the c reation of 4 × 3 × 7 × 5 = 420 differently placed grids with respect to the coordinate system (for each porosity) in the case of smaller zones, and 5 × 3 × 7 × 5 = 525 differently placed grids for larger zones (for each porosity). Along the width of the sample (crack front) an interval of the length l = 480 μm (corresponding approximately twelve times the lattice parameter for a porosity of 6 % or sixteen times that parameter for a porosity of 15 %) is chosen. In the half-space ahead of the crack front, the system of pores creating a new fracture surface is restricted to: (i) pores of coordinates x k , y k , z k the centers of which are lying in the direction of crack propagation; (ii) pores having a non-empty intersection with the plastic zone region ahead of the crack front. These conditions can be written as 0 k x  , z , k k k r l r  − + and ( ) 2 2 2 k k pz k x y R r +  + and only the pores, satisfying these conditions form a set F of all “usa ble ” spheres. A.2. Finding the shape of the crack inside the plastic zone The basic (primary) crack is a half-plane in the half space x ≤ 0, which is followed by a curved (tortuous) part inside the plastic zone. Here, the crack is modeled in xy sections ( z = const.) as a zig-zag polyline, formed by segments connecting the virtual shape for a constant projection of the distance from the crack front ( x = const). A disordered set of spherical pores that are a part of tortuous crack surface inside the plastic zone is denoted M (while  M F ) and characterized by two parameters x M and r M . The set is formed iteratively so that the initial state is an empty set with parameters x M = 0 and r M = 0. In the i -th step, a pore X with central coordinates ( x i , y i , z i ) and ) min( x | i M x     X X M F and min( | ) i i M r x y +     X X 2 2 M F applies, is added to the set M . The parameters of that new set are determined by M max( | ) i x x =  X M and M max( | ) i i r x y = +  X 2 2 M . This procedure is repeated until there is still a pore in the set F satisfying these conditions. In this way, all pores that are part of the crack surface inside the plastic zone are included in the set M . Thus, the positions of the bases of spherical canopies to which the zig-zag crack surface is connected are found. In each such defined plane ( x = const) the segmented polyline is constructed. Inside the pores, the segments are parallel with the basic plane xz (these segments lie within the bases of spherical canopies). The plane surface segments inside the pores are then substituted by spherical canopies simulating void patterns (dimples) on the real fracture surfaces. For a given z coordinate outside the pores, the positions of intersections are calculated from points on the neighboring planes which were determined by the previous procedure. In this way, a three-dimensional matrix is created, where the coordinates x and z correspond to the vertices of rectangles and the “height” coordinate y is related either to a pore or to the position determined by the linear interpolation between the pores. The tortuous fracture surface defined by that matrix is composed of rectangular segments simulating planar slip systems causing the shear fracture of inter-void ligaments inside the plastic zone.