Mechanical Metamaterial Unit Cell

Context

Mechanical metamaterials, also called architected materials, get their properties from their geometry as much as from the material they are made of. In the last part of the 2.002 project, each team designs a unit cell, has it printed at micrometer scale, and measures how it behaves, using the tools of nanomechanics: measuring mechanical properties of structures that are far too small for a standard testing machine.

Design

Starting from my own cell. Each of us designed a unit cell on our own first. Mine is a cube frame with crossing diagonal struts through its faces and center. In SolidWorks, a 5% compression predicted an effective modulus of about 174–202 MPa for the single cell, depending on loading direction, and about 346–398 MPa for a tessellation.

My individual design: a cube frame with crossing diagonal struts.

Combining designs. As a group, we took elements from each member’s design to arrive at one final unit cell. It is a star-shaped lattice of intersecting struts that tie together at shared nodes. It is mostly open space, so the question is how much stiffness the strut layout can carry. The cell was tessellated into a block of repeating cells for printing.

Predicting stiffness with FEA

Before printing, we ran linear compression simulations in SolidWorks to estimate the effective Young’s modulus, E*. Each model was compressed to 3% strain (kept in the linear regime), with the bottom face fixed to match the experiment. The effective modulus is the average stress on the loaded face divided by the applied strain.

• Single unit cell: E* ≈ 523 MPa
• 2×2×2 tessellation: E* ≈ 510 MPa

The tessellation is the better prediction of the printed block, since it includes the load sharing between neighboring cells.

Fabrication and measurement

The samples were printed by two-photon lithography, which uses ultrafast laser pulses to polymerize a photosensitive resin along a voxel only about 200 nm across: effectively the highest-resolution 3D printing available. Before testing, we measured the printed sample in an SEM image using Fiji, calibrating against the scale bar, and corrected the apparent vertical dimensions for the viewing angle. The block measures about 145 µm wide and 143 µm tall, with struts about 5.4 µm thick.

SEM image of the printed sample before compression (scale bar 100 µm).

Testing in situ

The compression test ran inside the SEM using a nanoindenter, an instrument with nanometer and micronewton precision. A flat punch presses down on the sample while the load–displacement data and a video are recorded together, so every feature of the load curve can be matched to what the structure is doing. Dividing load by the sample footprint, and displacement by its height, gives engineering stress and strain, from which come the effective modulus (steepest linear slope), yield strength (0.2% offset) and toughness (area under the curve).

The sample under the indenter after compression (scale bar 20 µm).

Outcome

The sample showed the behavior the handout describes: a toe region as the punch settles, a linear small-strain regime, yielding, and permanent deformation after unloading. From the stress–strain data we measured:

• E* — effective modulus, 637 MPa (the steepest linear slope over a 1% strain window)
• σ_y — yield strength, 14.3 MPa (0.2% offset)
• W — toughness, 1.92 MJ/m³ (area under the loading curve to 10% strain)
• E*/E_s — normalized stiffness, 0.372, against a monolithic pillar of the same polymer with E_s = 1,713 MPa

Engineering stress–strain curves for our metamaterial and the tetrakaidecahedron reference, with the FEM stiffness (dashed) and the experimental stiffness.

Our FEM estimate of 509.5 MPa came in about 20% below the measured 637 MPa. The likely reasons are the idealized CAD geometry, printing defects, uncertainty in the measured dimensions, and the fact that the simulation was linear elastic while the experiment included yielding.

von Mises stress on the unit cell from the SolidWorks FEA.