science.abn1459

Supplementary

Materials

for

Growth rules for irregular architected materials with programmable properties

Ke Liu, Rachel Sun, Chiara Daraio

Corresponding author

Chiara Daraio, daraio@caltech.edu.

Sci

ence

(

202

)

DOI:

10.1126/sci

ence

abn1459

The

PDF

file

includes:

Materials and Methods

Supplementary Text

Figs. S1 to S9

Other Supplementary Material for this manuscript includes the following:

Movies S1 to S

Materials and Methods

Sample

fabrication

The physical samples are 3D printed using the NinjaTek Semiflex filament (Fenner Inc.) on

CraftBot

Plus FFF 3D printers. Each sample is of size 100mm

100mm

15mm. We measured

the linear elastic properties of the Semiflex material, and obtained its Young’s modulus as

퐸퐸

푆푆

52.53

MPa, and Poisson’s ratio as

휈휈

푆푆

= 0.46

. The 3D model is printed by the Sintrat

ec Kit SLS

printer (Sintratec AG)

using Nylon material.

FEM

simulation

The mechanical properties of the 2D SVE samples are evaluated using custom

homogenization code written in Matlab, based on square-shaped Q4 elements, using periodic

boundary condition, assuming plane stress. Direct simulation on the 40×40 samples are

performed without periodic boundary condition. Each building block is mapped to a 20×20

pixelated discretization. The solid parts are modeled as linear elastic material with properties of

the Semiflex material. The voids are modeled as a very soft material (

퐸퐸

푉푉

= 0.001퐸퐸

푆푆

휈휈

푉푉

= 0

), a

technique that is often used in the field of topology optimization. The examples in fig. 5 are

performed in Comsol using second order triangular elements.

The

mechanical properties of the 3D samples are evaluated using custom homogenization

code written in Matlab, based on frame elements, with 3 translational and 3 rotational degrees of

freedom at each node, using periodic boundary condition.

From the homogeniz

ation analyses, we obtain the material’s effective elasticity tensor

푪푪

which describes the material’s infinitesimal response in all directions. The commonly used

material properties, such as the Young’s modulus and Poisson’s ratio, are functions of the

elasticity tensor. Let

푺푺 = 푪푪

−1

as the compliance tensor, the inverse of

푪푪

. Denote

흈흈

and

휺휺

as the

stress and strain tensor, respectively. If we define

퐸퐸

푥푥

퐸퐸

푦푦

, and

퐸퐸

푧푧

as the Young’s moduli in the

푥푥

푦푦

, and

푧푧

directions, respectively, they are given by:

퐸퐸

푥푥

휎휎

휖휖

푆푆

1111

, 퐸퐸

푦푦

휎휎

휖휖

푆푆

2222

, 퐸퐸

푧푧

휎휎

휖휖

푆푆

3333

(1)

The average Young’s modulus used in this article is defined as the average of the Young’s

moduli in different directions. For 2D:

퐸퐸

푎푎푎푎푎푎

�

퐸퐸

푥푥

+ 퐸퐸

푦푦

�

(2)

For 3D:

퐸퐸

푎푎푎푎푎푎

�

퐸퐸

푥푥

+ 퐸퐸

푦푦

+ 퐸퐸

푧푧

�

(3)

Similarly, the Poisson’s ratio along different

directions are defined by

휈휈

푥푥

푦푦

= −

휀휀

= −

푆푆

2211

푆푆

1111

, 휈휈

푦푦

푥푥

= −

휀휀

= −

푆푆

1122

푆푆

2222

휈휈

푦푦

푧푧

= −

휀휀

= −

푆푆

3322

푆푆

2222

, 휈휈

푧푧

푦푦

= −

휀휀

= −

푆푆

2233

푆푆

3333

휈휈

푥푥

푧푧

= −

휀휀

= −

푆푆

3311

푆푆

1111

, 휈휈

푧푧

푥푥

= −

휀휀

= −

푆푆

1133

푆푆

3333

(4)

The average Poisson’s ratio used in this article is defined as the average of the Poisson’s ratio in

all relevant directions

. For 2D:

휈휈

푎푎푎푎푎푎

�

휈휈

푥푥

푦푦

+ 휈휈

푦푦

푥푥

�

(5)

For 3D:

휈휈

푎푎푎푎푎푎

�

휈휈

푥푥

푦푦

+ 휈휈

푦푦

푥푥

+ 휈휈

푦푦

푧푧

+ 휈휈

푧푧

푦푦

+ 휈휈

푥푥

푧푧

+ 휈휈

푧푧

푥푥

�

(6)

The shear modulus of a 2D material is obtained by:

퐺퐺 =

푆푆

1212

(7)

For 3D, we can define shear moduli for different directions as:

퐺퐺

푥푥

푦푦

푆푆

1212

, 퐺퐺

푦푦

푧푧

푆푆

2323

, 퐺퐺

푥푥

푧푧

푆푆

1313

(8)

Mechanical

testing

and

DIC

The physical samples are tested on a Instron E3000 universal testing station with a 500-

load cell. The experimental setup is shown in fig.

S3 and movie S2. The samples are placed in

between two custom made aluminum fixtures. The contact surfaces between the fixtures and the

samples are lubricated by silicone lubricant with Teflon

(DuPont, Inc.), to reduce friction. For

the constituent material (Semiflex), tensile tests are conducted on five dog

-bone samples (ASTM

D638 Type IV). The Digital Image Correlation (DIC) analysis uses the VIC-2D system by

Correlated Solution, Inc. The speckled patterns are sprayed onto the samples using an air brush.

The DIC software captures the structures as point clouds and records the displacement of each

point. A rectangular bounding box is defined by averaging the locations of points within a 10

mm strip, next to each side of the bounding box. The deformation of the bounding box in each

direction is used to compute the average strains of the material sample. The average s

trains in the

axial and lateral directions are then used to calculate the experimental Poisson’s ratio, for axial

strain approximately within the range of 0.005-0.015.

The

virtual growth program

The virtual growth program is written in C#, modified from the “Wave Function Collapse”

code published by Maxim Gumin (@mxgmn) on GitHub. Our modified version (2D) is also

uploaded to GitHub, which is accessible from:

https://github.com/Daraio

lab/Virtual_Growth_Program

. T

e resultant databases (fig.

4) can be accessed from

the same

link

. Details of the algorithm is presented in the Supplementary Text. The 3D version and

corresponding data is available upon request to the corresponding author.

Supplementary

Text

The

virtual growth program

The implemented version of virtual growth program in the current research

consists of three

main input elements, which serve as the fundamental blueprint for the generation of the

architected materials: (i) the basic building blocks, (ii)

the adjacency rules, and (iii)

the

frequency hints, or target probabilities, of the basic building blocks.

The basic building blocks are elemental geometries that can be combined to produce

complex structures, for example, the four basic building blocks shown in fig.

S1B (“T”-

shape,

“L”-

shape, “

−

”- shape, and “+”-shape). In 2D, each basic building block also has eight

orientations through reflections and rotations, but some of them are identical due to symmetry.

As shown in fig.

S1B

, there are four distinct orientations for the “T”

-shaped building block, four

for the “L”-shaped building block, two for the “

−

”-shaped building block, and only one for the

“+”-shaped building block. The adjacency rules define whether and how the basic building

blocks can pair with each other spatial

ly, as illustrated in fig. 1C. The frequency hints of the

basic building blocks influence how many times each building block would appear in the final

design, which probabilistically controls the topology of the final design.

The growth process happens on a squared network of nodes for tile placement (

fig.

Starting fro

m an initially random tile placement, the algorithm assigns to each node on the grid a

random basic building block, but adjacency rules must be satisfied locally. In each step, the

algorithm first finds the node with minimal nodal entropy. The nodal entropy

푆푆

푖푖

of node

푖푖

defined by

푆푆

푖푖

= � −푃푃

푗푗

log 푃푃

푗푗

푗푗∈Ω

푖푖

(S1

)

In Equation (

푖푖

refers to the set of building blocks that are admissible to node

푖푖

. When

one or more neighbors of node

푖푖

have already been assigned, the number of available building

blocks reduces due to the compatibility constraint imposed by the adjacency rules. The symbol

푃푃

푗푗

denotes the normalized probability of building block

푗푗 ∈ Ω

푖푖

to be chosen, such that

∑ 푃푃

푗푗

= 1

푗푗

. In cases when multiple

nodes have the same entropy, the algorithm randomly

chooses one of them. This applies also to the initialization step when all node

s are empty, and at

the highest nodal entropy. After a node is chosen, the algorithm assigns a random building block

to it, sampled from all admissible ones according to their probabilities. Next, the algorithm

updates the admissible sets and nodal entropy values of all remaining empty node

s, and repeats

this procedure until all node

s are assigned. This process is illu

strated in fig. S

1D and m

ovie S1.

Defects can form when the admissible set of building blocks for a node

becomes empty. This

happens when the algorithm cannot assign any building block that it is simultaneously

compatible with all neighboring node

s already assigned, according to the adjacency rules.

Because the microstructures are composed of predefined building blocks and constructed

following rules that guarantee local compatibility, the final designs have no disconnected parts,

with boundarie

s and interfaces precisely defined.

The minimal nodal entropy heuristic ensures that the system’s total entropy decays at the

lowest rate, and thus results in a smaller chance of defects. The node with minimal entropy

each step of “growth” is most likely to be a neighbor of the last assigned node

, because the

assignment of one node

reduces the number of admissible building blocks in its neighboring

node

Although the virtual growth program is most suitable to produce irregular microstructures,

it can also generate periodic ones. When the adjacency rules apply to opposite boundaries of a

finite grid, we obtain microstructure motifs that can be periodically tessellated. This is equivalent

as if the opposite boundaries of a finite grid become connected to each other, topological

forming a torus.

The frequency hints used to construct the databases in fig. 4 and 6 of the main text are

generated by the followi

ng procedure. Denoting the frequency hint

of each building block as

푤푤

푖푖

to have a relatively complete coverage of all possible combinations of frequency hints, we need

to uniformly sample the space of {

푤푤

푖푖

} constrained by

∑

푤푤

푖푖

= 1. 0

. The sampling algorithm

follows the symmetric Dirichlet distribution, with screening to ensure relatively evenly spaced

points, which leads to 180 different combinations of frequency hints. To uniformly sample points

on the hyperplane of

∑

푤푤

푖푖

= 1. 0

, we first draw

푛푛

reals

(푝푝

, 푝푝

, ... , 푝푝

푛푛

)

on the interval (0,1),

following a uniform distribution. Then we define intermediate variables

푞푞

푖푖

= − log

푝푝

푖푖

. Finally,

we let

푤푤

푖푖

= 푞푞

푖푖

∑

푞푞

푖푖

⁄

, so that

푤푤

has a symmetric Dirichlet distribution, which is equivalent to a

uniform distribution over the open standard

(푛푛 − 1)

simplex. After a new point is sampled, its

Euclidean distances to all existing points are calculated, and only when all distances are larger

than a prescribed value

푑푑

, this new point is accepted. This procedure leads to uniformly

distributed points on the hyperplane of

∑ 푤푤

푖푖

= 1.0

, with density controlled by the parameter

푑푑

Expansion of Property

Space

In plane elasticity, materials’ elastic properties are categorized into four different symmetry

classes: isotropic, tetragon

al, orthotropic, and fully anisotropic. These symmetry classes

characterize how a material responds to deformations in different spatial directions. For an

architected material with periodic microstructure, its material symmetries are directly related to

the geometric symmetry of its periodic unit cell. For example, a square-shaped unit cell with a

four-

fold symmetric pattern tessellates into a material whose elastic property is of tetragonal

symmetry. A tetragonal material behaves the same in the

푥푥

and

푦푦

directions, but not in all

directions. In examples shown in figs. 3 & 4 of the main text, although the microstructures

generated by the virtual growth program are irregular, their ensembles, i.e., the bulk materials,

exhibit elastic responses that are clo

se to the tetragonal symmetry. This is due to both the choice

of a squared grid and the limited selection and orientation of building block geometry. To

demonstrate the ability to control and expand material symmetries, we demonstrate two possible

approach

es: (i) fine

-tuning the building block frequencies for different orientations, and (ii)

introducing new block geometries.

In all examples shown in figs. 3 & 4 of the main text, each possible orientation of a building

block was treated to have equal probability of occurrence. Therefore, the resultant microstructure

showed no directional preference over the

푥푥

and

푦푦

directions, which is the reason why the

materials display tetragonal symmetry. To expand the achievable material properties’ space, we

now assign unequal frequency hints to different orientations of the building blocks, as illustrated

in fig. S6. We separate the “T”

-shaped and “

−

”-shaped building blocks into two groups: one

with major load path along the

푥푥

direction, and the other with major load path along the

푦푦

direction. We gradually change the relative probability of the two groups from 50%-50% to

90%-10% (fig. S6A)

. For each of the new frequency

hints, the resultant material properties are

evaluated from 100 samples as before. Two representative samples are shown in fig. S6B. This

change of frequency hints tunes the material elastic properties from tetragonal to orthogonal

symmetry. As we increas

e the relative probability of the

푥푥

-major group, the Young’s modulus of

the generated material increases in

푥푥

direction, but decreases in

푦푦

direction (fig. S6C). The

oisson’s ratio also changes along different directions. As the relative probability of the

푥푥

-major

group increases,

휈휈

푦푦

푥푥

becomes more negative, while

휈휈

푥푥

푦푦

becomes less negative (

fig. S

). We

also observe that the distribution of

휈휈

푦푦

푥푥

over 100 samples become wider, while the distribution

휈휈

푥푥

푦푦

becomes narrower. Meanwhile, this ch

ange of frequency hints reduces the shear modulus

(

퐺퐺

) of the material in all directions (

fig. S6C), without inducing directional preference.

sing the four basic building blocks in fig. 1B leads to very weak directional Young’s

modulus in the 45° directions (fig. S6C). To improve the Young’s modulus in these directions,

we introduce three new basic building blocks: the “O”-

shape, the “V”-

shape, and the “\”-

shape

(fig. S6E

). We generate a new set of architected materials generated using all seven basic

building blocks and compare their properties to architected materials generated only using the

four initial blocks. Each set has 100 samples generated using the same inputs. The two sets are

chosen such that they exhibit similar Young’s modulus in the

푥푥

dire

ction. Two representative

samples from each set are shown in fig. S6F. We observe that adding more “O”-shaped building

blocks significantly improves the Young’s modulus in the 45° directions, without sacrificing the

Young’s modulus in the

푥푥

and

푦푦

directi

ons (fig. S6G). In addition, the new set of materials

displays a more isotropic shear modulus compared to materials obtained from only four building

block types. The presence of “O”-shaped building blocks plays an important role in increasing

the Poisson’s ratio, pushing both

휈휈

푥푥

푦푦

and

휈휈

푦푦

푥푥

from around 0.05 to around 0.65 (fig. S6H

Visualization

For the 3D irregular architected materials, we have created online interactive plot

s to

present the data: https://ob

servablehq.com/@mdeagen/cg09

, thanks to the help from Dr. Mike

Deagen (

mike.deagen@gmail.com

Fig. S1.

-C)

Illustration of the

“virtual growth”

process in 2D on a 5×5 grid of nodes. (A)

Four basic

building blocks (“T”

-shape, “L”

-shape, “

-”-shape, and “+”-

shape) and their possible orientations

for the “virtual growth” of materials

. (B) Examples of

local adjacency rules. Each rule specifies

how two

basic building blocks connect to each other. (C) At each step, the

node

with minimal

entropy is randomly assigned with

one of the

building blocks

from (A), according to the

frequency hints. This building block must be compatibility with its neighbors, specified by the

adjacency rules in (B)

. (D

-E) Emergence of defects

on 20×

20 grid of nodes

. The parameter

푁푁

denotes the ratio of the number of available building blocks over

total number of node

s. (D)

Typical defected designs for different

푁푁

. The gray dots indicate defects, where no available

building block can be assigned while satisfying compatibility constraints.

When

푁푁 = 1

, all

available building blocks must be perfectly fitted into all node

s, which is typically not possible

and thus leaves many defects.

(E)

For each combination of initial frequency hints, 1000 samples

of 20

20 grid of node

s are generated and counted. The density is define

d as the ratio of node

with no compatible building block to assign over all avai

lable node

s (i.e.

, 400). Different colors

refer to different combinations of initial frequency hints, as shown by the inset

. When

푁푁 = ∞

the frequency hints do not change due to assignment of building blocks, which reflects all

examples in the main text of this paper. When

푁푁

is finite, the assignment of a certain building

block type onto a node

reduces its frequency hint. When there are more “

−

” and “L”

-shaped

building blocks than “

” and “T”-

shaped building blocks in the environment, defects are more

likely to occur. This is

because the “

−

” and “L”

-shaped basic building blocks have less degrees

(or local connectivity).

Fig. S

Clusters of numerical and experimental data. Each data point refers to an independent sample.

Color code follows Fig.

in the main text.

The va

lues of

퐸퐸

푎푎푎푎푎푎

and

휈휈

푎푎푎푎푎푎

are taken as the

average of measured values in

푥푥

and

푦푦

directions.

Fig. S

Additional information from the experiments. (A) Differences between the prescribed frequency

hints and resultant probabilities of appearance.

(B) All stress

-strain curves of the 7 groups, each

containing 5 samples loaded in both

푥푥

and

푦푦

directions. The colored dots in the graphs

correspond to different frequency hints as shown in (A).

Fig. S

Snapshots from the experiment. (A) A sample of random architected material (from the same

group shown in fig. 3G of the main text) before load

ing. The sample is placed in between two

custom made aluminum blocks with shaped slots. (B) The deformed shape of the sample at 0.08

global (compressive) strain. Strong local nonlinear behavior is observed, resulting in contacts

between buckled elements an

d friction

Fig. S

Influence of the probability of appearance of the basic building blocks on different mechanical

properties. The horizontal axis of each diagram is the probability of appearance of the building

block shown as inset

within the diagra

m. Each color contains 100 samples generated using the

same growth rules.

Fig. S

Expansion of property space by altering inputs to the virtual growth program. (A) Change of

frequency hints of the “T” and “

−

” building blocks in different orientations

. The baseline

frequency hints are: 40% “T”, 40% “+”, 10% “

-”, 10% “L”. (B) Representative designs

generated by biased frequency hints. The left and right correspond to the 50%

-50% and 90%

10% groups, respectively. (C) Directional mechanical responses, nor

malized Young’s modulus

(

퐸퐸/퐸퐸

푆푆

) and normalized Shear modulus (

퐺퐺/퐺퐺

푆푆

) of the four groups in (A), each averaged over

100 samples.

퐺퐺

푆푆

denotes the shear modulus of the constituent material. (D) Distribution of

directional Poisson’s ratios (

휈휈

푦푦푥푥

and

휈휈

푥푥푦푦

) of the four groups in (A). (E) Three additional basic

building blocks and their possible orientations. They are referred to as the “V”

-shape, “\

”-shape,

and “O”

-shape, from top to bottom. (F) Representative designs from two groups of samples,

generated without (brown) and with (orange) the new building blocks, from left to right. The

frequency hints of the group with 4 basic building blocks: 3.03% “T”, 66.01% “+”, 8.57% “

−

”,

22.39% “L”. The frequency hints of the group with 7 basic building blocks: 4.73% “T”, 15.53%

“+”, 2.66% “

−

”, 6.29% “L”, 66.01% “O”, 1.26% “V”, 3.52% “

\”. (G) Directional mechanical

responses of the two groups. (H) Distribution of directional Poisson’s ratios (

휈휈

푦푦푥푥

and

휈휈

푥푥푦푦

) of the

two groups in (E).

Fig. S

Influenc

e of the probability of appearance of the basic building blocks on different mechanical

properties

for database #1 in fig. 6 of the main text. The horizontal axis of each diagram is the

probability of appearance of the corresponding building block. Each co

lor contains 100 samples

generated using the same growth rules.

Fig. S

Influence of the probability of appearance of the basic building blocks on different mechanical

properties for database #2 in fig. 6 of the main text. The horizontal axis of each diagram is the

probability of appearance of the corresponding building block. Each color contains 100 samples

generated using the same growth rules.