PSI - Issue 79

A. Della Rocca et al. / Procedia Structural Integrity 79 (2026) 475–484

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resembled the random variations observed in the effective stiffness, with intermediate nodes generally dominating the counts. In contrast, nodes with three or more connections (branch points) displayed a distinct trend: in fact, as both D and γ continued to increase within the Cahn–Hilliard model, the branch point count exhibited a clear decreasing trend. This reduction in high-connectivity junctions signals a structural evolution: the microstructure coarsens, thin struts disappear or merge, and the overall normalized connectivity decreases. This outcome suggests that the structure is becoming less topologically complex, even without a dramatic change in overall porosity (volume fraction). Ultimately, the normalized branch node density proved to be the most significant morphological indicator of connectivity and a strong correlation of effective stiffness.

Nodes with 1 connectivity

Nodes with 1 connectivity

1000 1100 1200

1000 1100 1200 1300

= 1 = 3 = 5 = 7 = 9

= 11 = 13 = 15 = 17 = 19

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D coefficient

Nodes with 2 connectivity

Nodes with 2 connectivity

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= 1 = 3 = 5 = 7 = 9

= 11 = 13 = 15 = 17 = 19

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D coefficient

Nodes with 3 connectivity

Nodes with 3 connectivity

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= 1 = 3 = 5 = 7 = 9

= 11 = 13 = 15 = 17 = 19

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D coefficient

Figure 5. Top, graph showing the number of end nodes for each D value and for different γ . (a) γ = 1, 3, 5, 7, 9. (b) γ = 11, 13, 15, 17, 19. Center, graphs showing intermediate nodes for each D value and for different γ . (c) γ = 1, 3, 5, 7, 9. (d) γ =11, 13, 15, 17, 19. Bottom, graphs showing 3 connectivity branch nodes for each D value and for different γ . (e) γ = 1, 3, 5, 7, 9. (f) γ =11, 13, 15, 17, 19.