www.sciencemag.org/content/358/6366/1072/suppl/DC1

Supplementary Materials for

Three-dimensional mechanical metamaterials with a twist

Tobias Frenzel, Muamer Kadic, Martin Wegener*

*Corresponding author. Email: martin.wegener@kit.edu

Published 24 November 2017, Science 358, 1072 (2017)

DOI: 10.1126/science.aan3456

This PDF file includes:

Supplementary Text

Figs. S1 to S7

References

Other Supplementary Material for this manuscript includes the following:

(available at www.sciencemag.org/cgi/content/full/358/6366/1072/DC1)

Movie S1

Materials and Methods

Sample fabrication

We fabricated the samples by 3D laser microprinting (Nanoscribe GmbH, Photonics Professional GT)

using a commercially available photoresist (IP-S, Nanoscribe GmbH). The liquid resist was

polymerized via multiphoton absorption using an Erbium fiber laser with center wavelength of

780 nm and 90 fs pulse length. A 25 × objective lens (numerical aperture NA = 0.8, Carl Zeiss)

which was dipped directly into the liquid photoresist was used to tightly focus the laser. Small

volumes of 300 µm × 300 µm × ℎ were scanned using galvanic mirrors (scan speed 𝑣 = 0.1 m/s)

and were stitched together using a mechanical stage. The height of the volume elements ℎ varied from

125 µm to 50 µm depending on the lattice constant 𝑎. Additionally, much smaller writing volumes

were used for overhanging parts such as the middle and top plates (compare Fig. 2). The underlying

3D models were created using the commercially available software package COMSOL Multiphysics

(COMSOL, Inc.) and further processed into machine code using the software Describe (Nanoscribe

GmbH). Unpolymerized resist was removed in a bath of mr-Dev 600, followed by a bath of acetone

and a supercritical-point-drying process.

Measurements

The measurements shown in Fig. 3 (also see Movie S1) were performed for 0.5%, 1%, and 1.5% axial

strain. The small differences suggest that deviations from linearity are small. Each measurement was

repeated 6 times. The shown points are averages. The resulting statistical errors are about the size of

the symbols in Fig. 3.

Supplementary Text

Micropolar continuum mechanics

In the static case, the micropolar equations of motion (18), reduce to

𝜕𝜎𝑖𝑖𝜕𝑥𝑖

= 0

𝜕𝑚𝑖𝑖

𝜕𝑥𝑖+ 𝜀𝑖𝑗𝑗𝜎𝑗𝑗 = 0.

Here, we again used the Einstein summation convention and assumed that the body force per unit

mass and the body couples per unit mass are zero. For the special case of cubic chiral materials

(pentagon ikosi-tetrahedral point group) relevant to the structure shown in Fig. 1C, the constitutive

equations are given by

𝜎𝑖𝑖 = 𝐶2𝜖𝑟𝑟𝛿𝑖𝑖 + 𝐶3𝜖𝑖𝑖 + 𝐶4𝜖𝑖𝑖 + 𝐶𝜖𝑖𝑖𝛿𝑖𝑖 + 𝐵2𝜑𝑟𝑟𝛿𝑖𝑖 + 𝐵3𝜑𝑖𝑖 + 𝐵4𝜑𝑖𝑖 + 𝐵𝜑𝑖𝑖𝛿𝑖𝑖

𝑚𝑖𝑖 = 𝐴2𝜑𝑟𝑟𝛿𝑖𝑖 + 𝐴3𝜑𝑖𝑖 + 𝐴4𝜑𝑖𝑖 + 𝐴𝜑𝑖𝑖𝛿𝑖𝑖 + 𝐵2𝜖𝑟𝑟𝛿𝑖𝑖 + 𝐵3𝜖𝑖𝑖 + 𝐵4𝜖𝑖𝑖 + 𝐵𝜖𝑖𝑖𝛿𝑖𝑖 .

For the results shown as red solid curves in Fig. 3, we solved these equations numerically by using the

partial differential equation (PDE) mode of the commercial software package COMSOL Multiphysics

(COMSOL, Inc.). Therefore, the weak form of the equations of motion has been implemented

according to

� �𝜎𝑖𝑖 �𝜕𝑢𝑖𝜕𝑥𝑖

− 𝜀𝑖𝑖𝑖𝜙𝑖�+ 𝑚𝑖𝑖𝜑𝑖𝑖�d𝑉 = � �𝜎𝑖𝑖𝑛𝑖𝑢𝑖 + 𝑚𝑖𝑖𝑛𝑖𝜙𝑖�d𝐴𝜕𝜕

.𝜕

The left-hand side is a volume integral, the right-hand side the integral over the corresponding surface.

𝑛𝑖 denotes the components of the exterior unit normal to the surface of the body. As to the boundary

conditions, we fixed one end face of a cuboid with volume 𝐿 × 𝐿 × 2𝐿 and pushed along the 𝑧-

direction onto the opposite facet using sliding boundary conditions. The following effective

micropolar material parameters were used: 𝐶2 = −6.14 MPa, 𝐶3 = 97.41 GPa, 𝐶4 = −97.37 GPa,

𝐴2 = 0.42 N, 𝐴3 = −1.85 N, 𝐴4 = 3.11 N, 𝐵4 = −7500 N/m, 𝐶 = 𝐵 = 𝐵2 = 𝐵3 = 𝐴 = 0. This

choice is not necessarily unique. However, the reasonable agreement with experiment (Fig. 3) shows

that the observed behavior can be mapped onto micropolar continuum mechanics. In sharp contrast,

ordinary Cauchy continuum mechanics for any parameters predicts strictly zero twist angle and strictly

constant Hooke’s spring constant or constant effective Young’s modulus versus 𝐿. In Fig. 3, we plot

these quantities versus 𝐿/𝑎 with 𝑎 = 500 µm to allow for a direct comparison with experiment.

Example raw data are shown in Fig. 4B.

According to (18), a characteristic length scale 𝑙c related to chirality follows from the elements of the

generalized elasticity tensors as

𝑙c = �𝐴2 + 𝐴3 + 𝐴4𝐶2 + 𝐶3 + 𝐶4

� (𝐵2 + 𝐵3 + 𝐵4)2(𝐴2 + 𝐴3 + 𝐴4)(𝐶2 + 𝐶3 + 𝐶4)

1 − 𝐶2𝐶2 + 𝐶3 + 𝐶4

.

For the above parameters, we obtain 𝑙c = 187 µm. This scale is roughly comparable to the size of the

unit cell, 𝑎 = 500 µm for 𝑁 = 1. However, this scale is much smaller than the one over which the

macroscopic twist effect decays: As shown Fig. 3A, the twist per axial strain decreases by a factor of

about two from 𝑁 = 1 to 𝑁 = 5, corresponding to a length scale of 5 × 𝑎 = 2.5 mm ≈ 13 × 𝑙c (also

see Fig. S5). Therefore, the characteristic length scale 𝑙𝑐 should be interpreted as connected to the

microscopic mechanism inside of the unit cell but it does not directly reflect the macroscopic

metamaterial behavior.

Static finite-element calculations

For the static finite-element calculations of the microstructures shown as crosses in Fig. 3, we solved

the ordinary Cauchy continuum mechanics equations for a microstructured constituent material

(compare Fig. 1C) with Young’s modulus 𝐸 = 2.6 GPa and Poisson’s ratio 𝜈 = 0.4 by using the

commercial software package COMSOL Multiphysics (MUMPS solver). We neglected geometrical

nonlinearities and assumed a strictly linear elastic behavior. We have validated this assumption by

exemplary calculations including geometrical nonlinearities (Fig. S3). In order to reduce

computational effort, only the bottom half of the experimental samples shown in Fig. 2, was actually

calculated. This reduces the number of unit cells to only 𝑁 × 𝑁 × 2𝑁 for the finite-element

calculations. To allow for a direct comparison with the experimental data in Fig. 3, the following

boundary conditions were implemented: one side of the sample was fixed to a rigid substrate; on the

other side, a square plate of side length 𝐿 and thickness 10 µm made of the same constituent material

as the sample was attached. Sliding boundary conditions were applied to this plate such that constant

forces and displacements in the 𝑧-direction were prescribed. Thereby, the corresponding the 𝑥- and 𝑦-

components of force and displacement field were kept free, which allowed for rotations. The twist

angle divided by axial strain was directly derived from the calculated data. The effective Young’s

modulus was obtained from the sample Hooke’ spring constant multiplied by the sample height, 4𝐿,

and divided by the sample cross section, 𝐿2. Geometrical parameters (compare Fig. 1C) were:

𝐿 = 500 µm, 𝑎 = 𝐿𝑁

, 𝑑 = 0.06 𝑎, 𝑏 = √2 𝑑, 𝑟1 = 0.32 𝑎, and 𝑟2 = 0.4 𝑎. For the chiral structures, we

chose the maximum possible angle allowed by geometry 𝛿 = asin�√2�𝑟2−

𝑑2�

𝑎−𝑏� ≈ 35°. For the achiral

structures, we chose 𝛿 = 0. Example raw data are shown in Fig. 4A.

Band structure and eigenmode calculations

We calculated the phonon band structures 𝜔(𝒌) of the chiral mechanical metamaterials by solving the

eigenvalue problem for a metamaterial crystal with unit cell according to Fig. 1C assuming Bloch

periodic boundary conditions. We used the commercial software package COMSOL Multiphysics

(MUMPS solver). As for the static calculations, we assumed a linearly elastic constituent material with

Young’s modulus 𝐸 = 2.6 GPa and Poisson’s ratio 𝜈 = 0.4. In addition, we assumed a mass density of

𝜌 = 1.15 g cm−3 and a lattice constant of 𝑎 = 500 µm (compare Fig. 2A). However, due to the

scalability of the continuum-mechanics equations of motion, the results can easily be scaled to other

lattice constants, Young’s moduli, and mass densities. Therefore, the frequency 𝑓 (see right-hand side

vertical scale) was converted to 𝑎/𝜆 (see left-hand side vertical scale), with the lattice constant 𝑎 and

the wavelength 𝜆 of the longitudinal mode, given by 𝑐/𝑓, with the phase velocity of the constituent

material longitudinal mode 𝑐 = � 𝐸(1−𝜈)𝜌(1−2𝑣)(1+𝜈)

.

As usual, the phonon band structure (see Fig. S6) refers to a fictitious infinitely extended periodic

crystal. In the true long-wavelength limit of ordinary Cauchy continuum mechanics, 𝜆/𝑎 ≫ 1, in

analogy to optical activity in optics (if properly considering dispersion), the two transverse modes

become degenerate and the mechanical twist vanishes. In contrast, towards the middle of the first

Brillouin zone at |𝒌| = 12𝜋𝑎, where the wavelength 𝜆 = |𝒌|/(2𝜋) is roughly four times the lattice

constant 𝑎, the degeneracy of the transverse acoustic modes (i.e., the two lowest-frequency bands) is

lifted and the corresponding phase velocities 𝜔/|𝒌| are different by nearly 5% when going along the

ΓX-direction. The corresponding eigenmodes exhibit left- and right-handed circular polarization,

respectively (see Fig. S7).

Supplementary Figs. S1-S7:

Figure S1: Illustration of twist mechanism. (A) Side view onto one unit cell (Fig. 1C). (B) Top view

(also see coordinate systems). The modulus of the local displacement vector (obtained from finite-

element calculations, see Methods) is shown on a false-color scale. We push onto the unit cell along

the negative 𝑧-axis with sliding boundary conditions at the top and bottom facets parallel to the 𝑥𝑦-

plane. The arrows aid our discussion: 1. Upon uniaxial loading, the arms connecting the corners with

the rings move downwards. 2. This motion leads to a rotation of the rings. 3. This rotation exerts

forces onto the corners within the 𝑥𝑦-plane, resulting in an overall twist of the unit cell around the

pushing axis.

Note that the center of mass of the depicted individual unit cell does not move at all in the 𝑥𝑦-plane

(see panel B). Therefore, if the unit cells within a metamaterial were not connected laterally, each unit

cell would rotate individually as depicted, but the overall twist of the metamaterial sample would be

strictly zero. Thus, sufficient lateral coupling of the unit cells is crucial to obtain a macroscopic twist

effect (Fig. 3A). This coupling also leads to the stiffening of a bar composed of 𝑁 × 𝑁 × 2𝑁 unit cells

versus 𝑁 (Fig. 3B) associated to a finite characteristic length scale.

Figure S2: Parameter variation. We have repeated the calculations shown as crosses in Fig. 3 for

𝑁 = 𝐿/𝑎 = 1 and the “standard” parameters as in Fig. 3 (highlighted in red here), except for the

geometrical parameters varied as indicated in panels (A)-(C) (compare Fig. 1C). Obeying fabrication

constraints, the standard parameters chosen in Fig. 3 are optimal in the sense of maximizing the twist

angle per axial strain. (A) variation of 𝑟2/𝑎 , (B) variation of 𝛿, (C) variation of 𝑑/𝑎. The relative

inner radius 𝑟1/𝑎 is not critical in the static case. It does, however influence the band structure (Fig.

S6) in the limit 𝑟1 → 0. Within reasonable bounds, the parameter 𝑏/𝑎 is not critical either.

Figure S3: Geometrical nonlinearities. (A) Calculated twist angle per axial strain as in the finite-

element microstructure calculations shown by the crosses in Fig. 3A for 𝑁 = 1, 2, 3, however, versus

the applied axial strain, while accounting for geometrical nonlinearities. (B) Effective Young’s

modulus versus applied axial strain. From these data it becomes clear that the relative deviations from

the linear regime are below 10% for strains around 1%. This finding is in agreement with the

experimental data (see three different symbols in Fig. 3). Moreover, experiment and theory also agree

in that the twist per axial strain slightly increases with axial strain, whereas the effective Young’s

modulus slightly decreases versus axial strain.

Figure S4: Alternative metamaterial unit cells. Using finite-element calculations (see Methods), we

considered alternative metamaterial blueprints based on the following unit cells. We used the

constituent material Young’s modulus 𝐸 = 2.6 GPa and Poisson’s ratio 𝜈 = 0.4. (A) As Fig. 1C,

repeated here only for convenience. (B) For 𝑁 = 1, this unit cell, which was suggested in (20), yields

twist angles per axial strain even larger than our blueprint (Fig. 3A). However, when going from

𝑁 = 1 to 𝑁 = 5, the twist per axial strain decreases by one order of magnitude, indicating an

undesirably small effective characteristic length scale. For our blueprint (Fig. 1C), the corresponding

decrease is merely a factor of two (Fig. 3A). (C) and (D) These two blueprints exhibit effects

comparable to that shown in (A), however, they are more demanding to fabricate. In particular, they

contain more overhanging structures during the 3D laser writing process. Furthermore, these non-

cubic unit cells lead to non-straight sample edges.

The unit cell and the metamaterial scaling versus 𝑁 could, in principle, be designed more rigorously

by using numerical topology optimization. However, one should be aware that the 3D forward

problem is already computationally demanding as one must not use periodic boundary conditions

(which would lead to zero twist angle per axial strain in the static case). One rather needs to simulate

the overall 3D metamaterial bar composed of 𝑁 × 𝑁 × 2𝑁 unit cells for each parameter combination.

Figure S5: Asymptotic behavior within micropolar continuum mechanics. The data underlying the

red curve are the same as those for the solid red curve in Fig. 3A, however, plotted up to larger values

of 𝐿/𝑎 (with 𝑎 = 500 µm) and plotted on a double-logarithmic scale. The upper horizontal scale is in

absolute values of the sample size 𝐿. For comparison, the straight blue line has a slope of −1. This

means that the ratio of twist angle and axial strain scales according to ∝ (𝐿/𝑎)−1, which is expected

intuitively (see main paper).

Figure S6: Band structure calculation. Frequency 𝑓 = 𝜔/(2𝜋) versus wave vector 𝒌 for the usual

tour along characteristic points of the simple-cubic Brillouin zone. Parameters correspond to the real-

space unit cell illustrated in Fig. 1C. The frequencies corresponding to the eigenmodes, which are

shown in Fig. S7, are highlighted in red. The lower panel shows the ratio of the phase velocity of the

first band, 𝑣1, and that of the second band, 𝑣2, versus wave vector.

Figure S7: Visualization of elastic eigenmodes. Visualization of three lowest-frequency eigenmodes

corresponding to the three points marked in the phonon band structure shown Fig. S6. Note that the

unit cell is cut differently here than in Fig. 1C. The two versions are connected by a shift by 𝑎/2 along

all three cubic axes. The red and blue arrows indicate the real and imaginary part, respectively, of the

displacement vector, spatially averaged over one metamaterial unit cell. The upper row shows an

oblique view, the lower row a view along the axis of wave propagation. If real and imaginary parts

have the same length, but include an angle of 90 degrees, the displacement vector rotates on a circle

versus time. Clearly, the two lowest-frequency transverse modes (A), (B) and (D), (E) have circular

polarizations with opposite sense of rotation with respect to the wave vector 𝒌 (black arrow). The next

higher-frequency mode (C) and (F) has longitudinal polarization.

Supplementary Movie S1:

The following link leads to an optical microscopy movie of the experiment for 𝑁 = 3. The left part of

the movie exhibits a bottom view of the sample onto the plate in between the left- and right-handed

part of the sample (compare Fig. 2C). The right part is a side view onto the same sample. Upon

pushing onto the sample, one can see a rotation around the pushing axis on the left and a compression

along the pushing axis on the right-hand side. The resulting rotation angle divided by the axial strain is

depicted in Fig. 3A for different samples. In contrast to Fig. 2C, all arrows are shown on the same

scale as the images in the background.

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