Pores and Stability

Cube Drilling Calculator — Surface/Volume + Stability Constraints

Cube side length, a Hole type Square (b × b) Rectangle (bw × bh)

Hole size, b (square b × b)

Hole width, bw Hole height, bh

Number of through-holes, N (along x-direction)

Assumption: holes are identical, do not overlap, and are placed on a simple grid on the yz-face.

Stability Inputs

Minimum remaining volume fraction, ρ*_min (e.g., 0.2) Minimum ligament thickness, t_min (e.g., 0.5)

What these mean:
• ρ* = V/V₀ is a quick “how much solid is left” indicator.
• t_y, t_z are the remaining wall thicknesses between holes (grid model).
• If ρ* < ρ*_min or t is below t_min, the structure is flagged “likely unstable”.

Footnote: What do ρ* (density indicator) and ligaments (tᵧ, t𝓏) mean?

This calculator is a geometry + first-order stability model. It estimates how much surface area you gain by drilling pores, but it also warns when the cube becomes mechanically weak.

1) Remaining solid fraction ρ*

We define ρ* as the fraction of solid material left after drilling: ρ* = V / V₀, where V₀ is the original cube volume (a³) and V is the remaining volume after drilling.

• ρ* = 1 → no holes (fully solid cube)
• ρ* ≈ 0.5 → about half the material removed
• ρ* → 0 → almost hollow, very fragile

The input ρ*_min is your chosen safety limit. If ρ* < ρ*_min, the code flags the design as “likely unstable” because there may not be enough solid framework to support loads.

2) Ligament thickness tᵧ and t𝓏 (the “walls” between holes)

When holes are placed on a grid on the yz-face, the cube is left with thin solid strips between neighboring holes. These strips are called ligaments (or “web thickness”).

If the holes are arranged as nᵧ × n𝓏 (enough to fit your chosen N), the pitch is: pᵧ = a/nᵧ,  p𝓏 = a/n𝓏. For a rectangular hole b_w × b_h, the remaining ligament thicknesses are:

tᵧ = pᵧ − b_w,   t𝓏 = p𝓏 − b_h

• If tᵧ or t𝓏 is small, the structure behaves like a thin frame → easy to crack/buckle.
• Increasing N reduces pitch → reduces t → lowers stability.

The input t_min is your minimum acceptable wall thickness (set it based on your material and fabrication). If tᵧ < t_min or t𝓏 < t_min, the code flags “likely unstable”.

3) Why we need both ρ* and t

• ρ* tells you how much total material remains (global strength indicator).
• tᵧ, t𝓏 tell you how thin the local connecting walls are (local failure indicator).

Practical note: This model is an engineering “screening tool”. Real stability also depends on material (E, σ_y, fracture toughness), hole-edge stress concentration, and loading direction. But using ρ* and t already prevents unrealistic “maximum surface area” designs that would collapse.