Structure, not composition, does the work
A block of titanium has one stiffness. Carve that same titanium into a lattice and you can dial its effective stiffness across two orders of magnitude — without changing the alloy at all.
That is the whole promise of architected (or "meta-") materials: the unit cell — the smallest repeating arrangement of struts, walls, or surfaces — becomes the design variable. Properties stop being a fixed lookup from a datasheet and become something you engineer geometrically. Change the shape, spacing, and thickness of the cell, and stiffness, strength, energy absorption, and permeability all move with it.
A field guide to lattices
Almost everything you'll meet falls into four buckets. The bucket largely decides how the cell deforms — which is the single biggest clue to its mechanical behavior.
Strut / truss
Networks of beams meeting at nodes. The node count per joint (connectivity) is what makes these either stiff or springy.
TPMS — sheet
A single smooth surface, given thickness, that splits space into two channels. Curvature everywhere — no stress-raising corners.
TPMS — solid / skeletal
The skeleton of the surface, thickened into a solid strut-like network. Smooth like a TPMS, but load paths behave more like a truss.
Stochastic
Randomized cells with no single repeating unit. Great for grading and for mimicking bone; more variable, harder to predict cleanly.
Everything hangs on relative density
If you learn one number, learn this one. Relative density is the fraction of the cell's volume that is actually solid:
Nearly every mechanical property is a power law in ρ̄. Double the solid you pack in and stiffness might quadruple — or merely double — depending entirely on how the cell carries load. That "how" is the next section, and it's the difference between a good design and a wasteful one.
Bending vs. stretching
Picture the joints of a lattice replaced with frictionless pins. Push on it. One of two things happens — and it changes everything downstream.
Bending-dominated
Struts rotate at the joints and bow. Compliant, forgiving, big flat plateau — excellent for absorbing energy.
Stretching-dominated
Bracing forces struts to carry axial load. Stiff and strong for its weight — the choice when you want structural efficiency.
The formal test is Maxwell's criterion: count struts b and joints j in the cell. If the frame has enough bracing to stay rigid as a pin-jointed truss, it stretches; if not, it bends.
Gibson–Ashby scaling laws
Gibson and Ashby distilled all of this into power laws. They are approximate, but they are the back-of-the-envelope tools that let you size a lattice in seconds. Two matter most — stiffness and strength:
The exponents are set by the deformation mode from Section 04 — this is where bending vs. stretching cashes out in numbers:
A lower exponent is a steeper payoff: at low density, a stretch-dominated cell (n≈1) is far stiffer than a bend-dominated one (n≈2) at the same weight. Watch it happen:
Which knob moves which property
Relative density and topology are the big two, but a handful of other parameters give you finer control. Here's the cheat sheet — hover any cell for the mechanism behind it.
| Parameter ↓ Property → | Stiffness E* | Strength σ* | Energy absorb. | Buckling resist. | Fatigue life | Perme- ability |
|---|
How the constants are actually found: homogenization
The scaling laws give you ρ̄n, but the constants C1, C2 — and the full directional behavior — come from homogenization. The idea is simple even if the machinery isn't: replace the complicated porous cell with an equivalent solid block that behaves the same at a distance.
Formally, you relate the volume-averaged stress to the volume-averaged strain through an effective stiffness tensor:
In practice — this is the "virtual testing" recipe you'll run in the tools below:
Build one representative unit cell (RUC)
The smallest tile that captures the cell's symmetry. Mesh it cleanly — and make the mesh periodic, so opposite faces match.
Apply six unit strains with periodic boundaries
Three normal, three shear. Periodic BCs make the single cell behave as if it's buried in an infinite array — no fake free-surface effects.
Average the resulting stress
For each load case, volume-average the stress field. That gives you one column of the stiffness matrix.
Assemble the 6×6 stiffness tensor
Six load cases fill the full C*. From it you read off E*, shear moduli, Poisson ratios, and — crucially — the anisotropy.
| C₁₁ | C₁₂ | C₁₂ | 0 | 0 | 0 |
| C₁₂ | C₁₁ | C₁₂ | 0 | 0 | 0 |
| C₁₂ | C₁₂ | C₁₁ | 0 | 0 | 0 |
| 0 | 0 | 0 | C₄₄ | 0 | 0 |
| 0 | 0 | 0 | 0 | C₄₄ | 0 |
| 0 | 0 | 0 | 0 | 0 | C₄₄ |
Direction matters more than you'd think
A lattice measured stiff along Z can be a noodle at 45°. That directional stiffness surface — plot E as a function of loading direction and you get a lobed 3D shape, not a sphere — is often the difference between a part that works and one that surprises you in service. Octet trusses are strongly anisotropic; many sheet-TPMS (gyroid especially) are close to isotropic, which is a large part of why they're trusted in implants and pressure-loaded parts. When you orient a lattice, you're orienting its whole property surface with it — align the stiff axis with your primary load.
PI-TPMS — phase-intersected minimal surfaces
Once the fundamentals click, here's a technique from our own bench that's fully public. Standard TPMS give you smoothness but pin you to bending-ish behavior at low density. Phase-intersected TPMS (PI-TPMS) takes two (or more) copies of a TPMS field, phase-shifts them, and keeps only where they intersect. The intersections lay down strut-like reinforcement along the surface seams — you get TPMS smoothness with more truss-like, stretch-efficient load paths.
Two details make it practical:
Predictable density. Relative density scales with the square of the controlling radius parameter — ρ̄ ∝ r² — so density becomes a clean, monotonic knob instead of a fiddly threshold hunt.
Round struts, by construction. A naïve iso-threshold leaves lens-shaped, oval strut cross-sections that concentrate stress and print poorly. Swapping the threshold for a true Euclidean distance-to-surface gives genuinely cylindrical struts — better fatigue, better printability. We call it the distance-formula cylindricality fix, and it's open for anyone to use.
The point for you as a learner: TPMS isn't a fixed catalog. Once you can read the implicit field, you can compose new cells with properties the catalog doesn't have.
The GUI-first toolchain
You do not need to write code to build real intuition. Start clicking. Generate a cell, sweep the density, look at what changes. These are the click-and-drag tools — all free unless noted — ordered roughly generate → explore → analyze.
MSLattice
Free GUIThe friendliest starting point. A MATLAB app (runs standalone) for uniform and graded TPMS — set topology, relative density, cell size, grading, hybridization, sheet or solid, export STL. Best first hour you can spend.
Lattice_Karak
Open · GUIOpen-source GUI covering density grading, cell-size grading, hybridization, and hierarchical cells. Runs as a standalone app — no MATLAB license required.
LattGen
Open · GUIOpen-source with the raw code exposed, equation-driven grading, and a large library of minimal-surface functions if you want to reach past the usual gyroid/Schwarz set.
F13LD suite
Not a Robot · browserOur own browser-native, single-file tools — nothing to install, WebGL/WebGPU rendering, Manifold export. TPMS, Grain, Foam, Noise, Mesh (3MF), plus write-ups in Field Notes. Built exactly for this kind of parameter play.
LatticeRobot
Interactive indexAn interactive environment that indexes the real, empirical properties of lattices, textures, and metamaterials — mix base materials and geometries and see data-driven, optimized implicit cells. The fastest way to build a felt sense for the parameter → property map without printing a thing.
F13LD · Lab / Synth
Not a Robot · browserLab runs in-browser homogenization on a cell you built; Synth flips the problem — name the properties you want and it suggests the geometry (inverse design). Both stay in the click-and-drag world.
TPMS Designer
Open · GUIWhere generation meets analysis. Open-source MATLAB toolbox (standalone version too) that generates and characterizes TPMS — pulls out surface, morphology, and mechanical metrics you can feed into CAE.
Microgen
Python · when readyThe step past the GUIs. Open-source Python for RUC generation and periodic meshing (TPMS, octet, Voronoi), exporting solver-ready files for real FEA homogenization. Worth knowing exists for when you want the full 6×6.
If I were starting today
Get the theory from the source
Work through the free MIT OpenCourseWare course on Cellular Solids (Lorna Gibson) — full video lectures and notes. This is Gibson–Ashby, taught by the person who wrote it, with a strong bone/biomimicry thread.
Make a hundred cells
Open MSLattice or an F13LD tool and sweep. Change one parameter at a time — density, then topology, then grading — and build muscle memory for what each does to the shape.
Connect geometry to numbers
Come back to the scaling explorer above with real intent, then browse LatticeRobot to see how measured data tracks (or defies) the tidy power laws. This is where intuition sets.
Analyze one cell properly
Take a single cell through TPMS Designer or F13LD.Lab. Read its stiffness and anisotropy. Now the 6×6 matrix means something because it's your cell.
Print it and break it
Nothing calibrates like a real compression test. Print a few densities, crush them, plot force–displacement against the scaling laws. The gap between prediction and reality is the whole rest of the field.
Sources & open reading
- Gibson, L. J. & Ashby, M. F. — Cellular Solids: Structure and Properties, 2nd ed. The canonical text. Its theory is taught free via MIT OCW 3.054 (video + notes).
- MIT OpenCourseWare 3.054 — Cellular Solids: Structure, Properties and Applications (Lorna Gibson). Full course, free. ocw.mit.edu ↗
- Architected cellular materials review — mechanical properties toward fatigue-tolerant design; strong on the bending/stretching classification. ScienceDirect ↗
- Effect of Lattice Topology on Mechanical Properties — open-access study, 55 printed lattices across topology and density (modulus, yield, plateau, energy). A clean worked parameter sweep. PMC open access ↗
- Mechanical Homogenisation of TPMS Architectures — FE vs. Mechanics-of-Structure-Genome, using the open-source Microgen tool. Good on the virtual-testing recipe. PMC open access ↗
- Al-Ketan, O. & Abu Al-Rub, R. K. — MSLattice: free software for uniform and graded TPMS lattices. Wiley ↗