Supplementary Information
of:
Control of Mechanical and Fracture Properties in Two
-
phase Materials Reinforced by
Continuous, Irregular Networks
Tommaso Magrini*, Chelsea Fox, Adeline Wihardja, Athena Kolli, Chiara Daraio
*
Engineering and Applied
Science, California Institute of Technology, Pasadena, CA 91125, USA
*magrini@caltech.edu
,
*daraio@caltech.edu
Legend
Experimental Section
Supplementary
Figures
:
Figures
SI 1
-
SI
12
Supplementary Discussion 1
-
5
Supplementary References
Experimental Section
Sample Fabrication |
Samples were generated using the virtual growth algorithm described by
Liu et. al.
28
and further described in the Supplementary Information (Supplementary Discussion
1). The virtual growth algorithm provides a PNG file of the sample architecture, which is then
edited using Adobe Illustrator to smoothen all tile connections, ensuring the s
ame volume
fraction of reinforcing phase and matrix phase in each sample.
Finally,
each phase of the
sample
is extruded and converted into a separate STL file for printing
. The specimens are then printed
using a Polyjet printer Stratasys Objet500 Connex3,
that has a lateral resolution of 40
-
85 μm.
1
The reinforcing phase and matrix phases are printed from Stratasys® VeroWhite Polyjet Resin
and Stratasys® TangoBlack Polyjet Resin, respectively.
Mechanical Characterization
|
Uniaxial tension tests were performed on plate geometries of the
additively manufactured polymeric composites, with dimensions of 75 mm x 75 mm x 5 mm. An
Instron E3000 (Instron, USA) with a 5kN load cell was used to apply a small preload followed
by a qua
si
-
static tensile loading at a rate of 2 mm/min. The measured force and displacements
were then used to calculate the tensile engineering stress and strain. The experiments were
recorded using a Nikon D750 camera (Nikon, USA) with a Nikkor 120 mm f/4 lens
(Nikon,
USA) at a rate of 1 frame per second.
Digital Image Correlation |
The same camera setup was used to perform 2D digital image
correlation (DIC) on equivalent sets of samples. Samples were painted white using flat white
spray paint and then speckle
d using flat black spray paint such that each speckle was
approximately 0.1 to 0.3 mm in diameter and would take up approximately 3x3 to 5x5 pixels of
each image. VIC
-
2D digital image correlation software (Correlated Solutions, USA) was used to
calculate t
he displacements and the resulting Lagrangian strain fields across the different
substructures, using a subset size of 31 and a step size of 2, which captured the large global
deformation while allowing for sufficient resolution of the local deformation.
Supplementary Figures
Figure SI 1
Figure
SI
1
. Connectivity rules for (L), (T) and (
-
) tiles, with allowed connections (green shade)
and prevented connections (red shade).
Figure SI 2
Figure
SI
2. (a) Meso
-
structure dimension analysis using elliptic fit to find Xm (meso
-
structure
size)
,
performed by image analysis
.
2
(b) Meso
-
structure size distributions for (A) and (B)
architectures relating Xm dimension to concentration percent. Yellow, green
, cyan and blue
represent 4, 6, 8, and 10+ tiles meso
-
structures, respectively.
Figure SI 3
Figure
SI
3. (a,b) Meso
-
structure angle distribution in (A) and (B) networks (a and b
respectively).
Figure SI 4
Figure
SI
4. (a) Meso
-
structure distributions around (L)
-
dominated 4
-
node substructures. (b)
Substructure distributions around (T)
-
dominated architectures.
Figure SI 5
Figure
SI
5. (a) Engineering tensile strain
-
stress diagram plotted with opening angle of (L) tile as
tensile loading is applied. (b) Progression of (L) tile angle opening under tensile loading.
Figure SI 6
Figure
SI
6. Comparison of reinforcing network and compos
ite failure locations under tensile
loading. (a) (A) reinforcing network and (b) (A)
-
NRC showing the same failure locations. (c) (B)
reinforcing network and (d) (B)
-
NRC showing the same failure locations.
Figure SI 7
Figure
SI
7.
(a) Simple tension experiment performed on VeroWhite and TangoBlack specimens.
(b) Single edge notch tension (SENT) experiment performed on VeroWhite and TangoBlack
s
pecimens at a loading rate of 1mm/min. (c) Evolution of the crack position over time durin
g the
fracture of a VeroWhite
-
TangoBlack
-
VeroWhite specimen, confirming sequential events of
crack arrest at the interface (I and II).
The crack position has been retrieved by image analysis.
2
(d) Digital Image Correlation (DIC) maps of the e
yy
strain
during the test reported in (c). The
inset confirms the sequential formation of a plastic zone in the VeroWhite portions subject to
local yielding, as also observed in literature.
2,3
(e) Optical photographs during a controlled
fracture experiment (SENT geo
metry) of a network reinforced composite loaded at 1mm/min
(initial crack length/specimen width ~ 0.5). Sequential details display the instantaneous
propagation of a crack at small displacements internally to the first mesostructure, within the
TangoBlack
matrix (scale bar = 1 mm). Circular insets highlight the event of fracture arrest at the
first soft to hard interface and the nucleation of a void in a nearby mesostructure. (f) Optical
photographs highlighting the sequence of craze formation, fibril elong
ation and fracture within
the TangoBlack matrix phase, internally to a mesostructure.
Figure SI 8
Figure SI 8:
(a) Stress strain curves recorded on A
-
NRCs with a different reinforcement volume
fraction V
R
, ranging from 0.15 to 0.45. (b) Modulus of
Toughness (MOT) as a function of the
reinforcement volume fraction. (c) Optical photographs depicting the evolution of the fracture in
specimens with increasing reinforcement volume fraction 0.15 (left), 0.30 (center) and 0.45
(right).
Figure SI 9
Figure SI 9:
(Left) Optical photographs of composites with increasing reinforcement volume
fraction V
R
ranging from 0.15 to 0.45. The ligament thickness map
is obtained with the plugin
‘local thickness’
3
in the open
-
source image analysis software Fiji
2
and it
is
then
overlayed on the
images. (Right) Ligament thickness distribution for each specimen (bars), smoothing of the
distributions measured for each specimen (solid lines).
Figure SI
10
Figure
SI 10
:
Engineering tensile stress
-
strain diagrams
showing connectivity rule modification
effects. (a) Original (A) architecture (lower curve) comparison with first iteration (upper curve)
of connectivity rule modifications as shown in upper right corner. (b) Original (B) architecture
(lower curve) compari
son with first iteration (upper curve) of connectivity rule modifications as
shown in upper right corner.
Figure SI
11
Figure
SI 11
.
Variations in distribution of meso
-
structure sizes for (A)
-
NRCs and Mod
-
(A)
-
NRCs (left) and (B)
-
NRCs and Mod
-
(B)
-
NRCs
(right)
,
measured by image analysis
.
2
Yellow,
green, cyan and blue represent 4, 6, 8, and 10+ tiles meso
-
structures, respectively, before (white
bars) and after (red and blue bars) connectivity rule modifications.
Figure SI
12
Figure
SI 12
.
Images
(left) and digital image correlation maps (right) recorded at 0.5% strain of
(A)* and (B)* composites (top and bottom, respectively).
Supplementary Discussion
1 | The growth algorithm
The program
used to design the composites
iteratively grows irregular metamaterials over a
predefined area by selecting spatial sites, defined by two coordinates, and assigning tiles to these
coordinates
[Liu, Sun, Daraio, Science, 377(6609), 975
-
981 (2022)]
.
4
The entropy of each site,
defined
as the number of available connections that can still be formed, a number that ranges
from 1 to 4, is evaluated, and the sites with the lowest entropy are filled first. The algorithm then
randomly selects tiles based on their initial concentration or avail
ability, reported often as
‘frequency hints’ (Figure 1b, left). This parameter is provided as an input by the user. To ensure
a smooth growth without local defects or discontinuity between the reinforcing and matrix
phases, the connectivity of each tile is
governed by a set of connectivity rules, also known as
adjacency rules (Figure SI 1). As the structure grows and the sites are occupied, each remaining
available site can only be occupied by a limited number of tiles, arranged in a limited number of
rotat
ional configurations (Figure 1b, center). The growth algorithm then proceeds with the
random selection of one of the allowed tiles until it ensures the complete filling of the predefined
area (Figure 1b, right). For this study, a specific (L) to (L) connec
tion is not allowed, to prevent
the formation of secondary disconnected architectures (Figure SI 1).
Supplementary Discussion 2 | The Reinforcing Networks
Networks are mathematical models that describe systems of nodes connected by edges. The rules
that g
overn how nodes connect ultimately determine the network's characteristics, distinguishing
an irregular network, which follows a set of rules to form an irregular pattern, from a random
network, where nodes are connected randomly. Relying on the theoretica
l frameworks developed
for continuous random networks, that describe the relationship between networks architecture
and mechanical properties, we can characterize the reinforcing networks generated using the
virtual growth algorithm based on their average
coordination <R>, the mean number of
connections each node has, and their bridge length, defined as the distance between two nodes.
In this context, among the different models developed, covalent networks have been extensively
studied to describe amorphous
solids and glasses, composed by atoms with different
coordination, analogous to our system tiles. The average coordination of covalent networks <R>
can be measured through constraint counting, estimating the network's stability and identifying
floppy mode
s, independent deformations that occur with no cost in energy. The critical
coordination <R
C
> marks the transition between rigid and floppy behavior and can be used to
predict the mechanical behavior of the network.
5,6
Supplementary Discussion
3
| The
Mechanical Behavior of the Reinforcing Networks
To confirm this description of the failure mechanisms in (L)
-
dominated and (T)
-
dominated
architectures, we performed control experiments on the reinforcing
networks
and compared the
results with the
behaviour
of the composites.
(L)
-
dominated architectures: the experiments on the
reinforcing networks
consolidated our
description of a transition from a bending
-
to a stretching
-
dominated
behaviour
. As the strain
exceeded 20% (figure 3a, grey solid line and grey a
rrows, and Figure SI
6
) we observed the
sequential fracture of the reinforcing
network
ligaments. Finally, these experiments allowed us to
shed light on the role of the matrix during fracture. The reinforcing
networks
display a
significantly lower strength
than the composites: this suggests that the matrix
has a
key
role
during loading in resisting against the deformation of
the
meso
-
structures, requiring higher
forces and thus increasing the total amount of energy that is needed to cause composite failure
(Figure 3a, black solid line composite, grey solid line
reinforcing network
).
Nonetheless, the
matrix has a marginal role in influencing the trajectory of the cracks, that can almost perfectly
overlap between the composite and the
reinforcing network
.
(T)
-
dominated architectures:
To consolidate our description of their mechanical behavior, we
performed control experiments on the reinforcing
network
of (B) architectures.
A
s expected
, they
display a primarily stretching
-
dominated behavior, that features seque
ntial failure events at lower
strain values (Figure 3b, grey arrows, Figure SI
6
). As observed in the (L)
-
dominated
architectures, the crack trajectory can almost perfectly overlap between the composite and the
reinforcing network
.
Supplementary Discussi
on
4
| Iterative modification design process
With our approach, we display that without changing the shape or the volume fraction of the
constitutive tiles, and without any shape optimization, it is possible, simply by changing the rules
that govern the connectivity of tiles, to tune the mechanical p
roperties of these architected
materials. Although this study focused on improving the energy dissipated during fracture of the
architectures, the iterative process of modifying connectivity rules and characterizing the
mechanical response allowed us to id
entify which set of rules can be modified to increase or to
decrease various other mechanical properties. The two specific mechanical properties we
examined during the iterative process were strength and strain
-
to
-
failure, which combined to
give an improve
d fracture energy dissipation.
We began by removing the most prominent defect in each architecture. The tile distributions are
directly responsible for the frequency of the type of defect in each architecture, and the most
prominent defect is the one that
occurs most often in an architecture. For (A), that was certain
(L) to (L) connections, while for (B), certain (T) to (T) connections were the most problematic.
Although the modified architectures (A) and (B) were generated using the exact same frequency
hints as their original counterparts, the modification of the connectivity rules, and thus the
possible substructures that could be formed, influenced their final composition: the modified
designs (A_NO LL) and (B_NO TT) have a composition of 50% (T), 19%
(L), and 31% (
-
) and
68% (T), 22% (L), and 10% (
-
),
respectively.
We then performed the same mechanical testing and characterization of the modified (A) and (B)
composites as described for the original composites. For both composites, the removal of the
m
ost prominent defects resulted in an increase in the modulus of toughness, but the mechanism
behind the increase was different for (A) and (B) (Figure SI
7
). The increase in modulus of
toughness for (A) was due to an increase in strength, while the increas
e in modulus of toughness
for (B) was due to an increase in strain
-
to
-
failure. From this, we concluded that removal of (L) to
(L) connections is responsible for strength and that removal of (T) to (T) and (
-
) to (
-
)
connections is responsible for strain
-
to
-
failure.
To test this hypothesis, we then removed the (
-
) to (
-
) connections from (A) to improve its strain
-
to
-
failure (resulting in 37% (T), 58% (L), and 5% (
-
)), and the (L) to (L) connections from (B)
(resulting in 83% (T), 9% (L), and 8% (
-
)) to imp
rove its strength. We again performed the same
mechanical testing and characterization of the newly modified (A) and (B) architectures, and as
expected, observed that the removal of (
-
) to (
-
) connections improved the strain
-
to
-
failure
from
the original (A
) architecture, while removal of (L) to (L) connections improved the strength from
the original (B). However, unlike the first modification, the second modifications did not have as
great of an impact on the increase in modulus of toughness.
The improveme
nts in strength and strain
-
to
-
failure are due to the tile distributions and thus
meso
-
structure populations that result from the connectivity rules applied. Since
energy dissipation
during fracture
is optimized by a combination of strength and strain
-
to
-
fa
ilure, we then decided
to combine all the modulus of toughness improvements into a final set of connectivity rules that
were applied to the original (A) and (B) architectures.
Supplementary Discussion
5
| The Polydispersity Index (PDI)
Borrowing the con
cept from the field of polymer science, a measure of the polydispersity of a
polymer is the polydispersity index (PDI), defined as the ratio of Mw (weighted average mass)
over Mn (number average mass). The parallel between polymers and irregular architecte
d
materials is apparent in this case; each tile can be seen as one monomer, and each formed
substructure can be seen as one polymer chain. The total amount of tiles (or monomers) is fixed
by the total extension of the material and the final assembly will t
herefore be a collection of
differently sized
meso
-
structures (or polymer chains). In this context, we propose here that
comparing the PDI of each architecture can be a quick method to evaluate the homogeneity of
each architecture.
The PDI is calculated
following Equation 1:
⁄
(1)
In which
and
are given by Equation 2 and 3 respectively:
∑
∑
⁄
(2)
∑
∑
⁄
(3)
is the enclosed area of a n
th
meso
-
structure and
is the frequency (the number of
times) it gets measured.
As it is defined, the PDI is larger than 1.0 and the closer it is to 1.0, the more homogeneous the
architecture is. As a result of more stringent connectivity rules, that bias the growth of
archite
ctures capable of higher energy dissipation during fracture, we observed across the design
space an overall decrease of the PDI, suggesting that the modified architectures become more
homogeneous (Table 1).
Table 1:
Architecture #
(T)
(
-
)
(L)
PDI Original
(T)*
(
-
)*
(L)*
PDI Modified*
1
62
18
20
1.78
63
18
19
1.41
2
60
29
11
1.81
65
18
17
1.37
3
57
6
37
2.08
64
12
24
1.58
4
42
45
13
1.85
58
22
20
1.39
5
19
2
79
1.80
49
26
25
1.36
6 (A)
35
10
55
2.09
54
19
27
1.41
7
36
28
36
2.27
53
24
21
1.51
8 (B)
80
10
10
1.81
71
11
18
1.45
Supplementary
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