Topology and deformation mode shape the exponent; density sets the operating point
Pick a topology class and a relative density. The readout shows the idealized stiffness fraction E*/Es ≈ C(ρ*/ρs)n; the dot shows where it lands on the two scaling curves.
Topology class
Relative density
Predicted stiffness fraction
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E*/Es vs relative density (log–log)
n = 1 · stretch-dominated (ideal)n = 2 · bending-dominated (ideal)n = 1.85 · illustrative TPMS — shown when selected
Idealized textbook scaling (Gibson–Ashby; C taken as 1 for teaching): bending-dominated architectures follow n ≈ 2, stretch-dominated approach n ≈ 1. The intermediate TPMS exponent shown (n = 1.85) is illustrative rather than universal — a gyroid’s fitted scaling depends on sheet-based vs skeletal form, relative-density range, loading direction, base material, defects, and manufacturing route. Printed lattices deviate — strut defects, nodes, and surface roughness shift real exponents and strengths — so treat this as intuition, and verify structural parts on printed coupons. Not a structural design tool.