Control of Mechanical and Fracture Properties in Two‐Phase Materials Reinforced by Continuous, Irregular Networks - Advanced Materials - 2023 - Magrini - Control of Mechanical and Fracture Properties in Two‐Phase Materials Reinforced by.pdf

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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

Abstract

Composites with

high strength and high fracture resistance are desirable for structural and

protective applications. Most composites, however, suffer from poor damage tolerance and

are prone to unpredictable fractures. Understanding the behavior of materials with an

irre

gular reinforcement phase offers fundamental guidelines for tailoring their performance.

Here, we study the fracture nucleation and propagation in two phase composites, as a

function of the topology of their irregular microstructures. We use a stochastic a

lgorithm to

design the polymeric reinforcing network, achieving independent control of topology and

geometry of the microstructure. By tuning the local connectivity of isodense tiles and their

assembly into larger structures, we tailor the mechanical and f

racture properties of the

architected composites, at the local and global scale. Finally, combining different reinforcing

networks into a spatially determined meso

-

scale assembly, we demonstrate how the spatial

propagation of fractures in architected compo

site materials can be designed and controlled

a

priori

.

Introduction

Composite materials offer many advantages over traditional materials, such as being

lightweight while maintaining a high strength and stiffness,

[1,2]

but they suffer from lack of

tough

ness and poor damage tolerance.

[3

–

6]

One way to improve their crack response is to

tailor the reinforcing phase architecture.

[7

–

10]

Fiber reinforcements, for example, exploit crack

bridging between fibers for toughening.

Introducing fibers and other hi

gh

-

aspect

-

ratio

reinforcing elements in the design of composite materials, often leads to direction

-

dependent

mechanical properties and anisotropic fracture resistance.

[11]

Depending on the reinforcing

elements‟ alignment direction, composites can be eit

her toughened by high fracture energy

dissipative mechanisms, such as fiber bridging and fiber pullout, or be subject to

2

delamination fractures, which occur at the fiber

-

matrix interface.

[11

–

14]

On the contrary,

randomly distributed inclusions, which

primarily toughen the material through microcracking

and secondary crack formation, often lead to composite materials with isotropic fracture

properties.

[15

–

17]

Developing materials that use multiple toughening mechanisms, like

bridging, deflecting, or e

ven arresting the propagation of cracks, has potential to improve the

amount of absorbed fracture energy.

This was recently demonstrated in bioinspired

architected composites, where the internal microstructure is finely tailored to control crack

propagatio

n behavior.

[18,19]

The combination of multiple toughening mechanisms can also be

achieved by fabricating composite materials with irregular reinforcing networks

[20,21]

. Irregular

microstructures are common in biological structural materials

[22

–

25]

a

nd understanding their

behavior during loading and fracture is relevant for the design of architected materials with

tailored load

-

bearing performance. Irregular networks can control the fracture and

toughening behavior of materials through the creation of

meso

-

scale structures with different

dimensions and orientations that cause multiple fracture nucleation and propagation events.

Finally, reinforcing composites with irregular networks allows the creation of materials with

direction

-

independent mechanical

properties, a desirable feature in structural and load

-

bearing applications. Here, we describe how network coordination influences the global

mechanical properties of two

-

phase materials, like strength, stiffness and energy dissipated

during fracture, as

well as the role of local mechanisms on fracture nucleation and

propagation. Introducing desired irregular networks as composite reinforcement and

achieving a fine control over their assembly across multiple lengthscales, from the micro

-

to

the cm

-

scale, r

equires advances in both numerical design and manufacturing. In recent

work, machine

-

learning and data

-

driven approaches were used to computationally design

hierarchical architected materials.

[26]

Here,

we employ algorithms that “grow” regular and

irregu

lar networks

[27]

for composite design and use multi

-

material additive manufacturing

processes for fabrication.

Design of irregular reinforcement

To design the stiff reinforcement phase of our two

-

phase composites, we utilized the virtual

growth algorit

hm

(Supplementary Discussion 1

), which t

e

ssellates a set of bimaterial tiles on

a discretized spatial grid, following a set of adjacency rules

22

. We used a combination of 2

-

coordinated tiles ([L] and [

-

]) and 3

-

coordinated tiles ([T]) and ensured that each

tile had the

same volume fraction of stiff reinforcing phase and soft matrix phase

(Figure 1a, left)

. We

combined these tiles to generate composites with a stiff reinforcing random network (white)

and a soft elastomeric matrix (black)

(Figure 1a, right)

.

The virtual growth algorithm ensures

continuity between the two phases through modifiable connectivity rules

(Figure SI 1)

.

Depending on the relative composition of 2

-

and 3

-

coordinated tiles, the virtual growth

algorithm creates various composites with the same volume fraction of reinforcement, but a

large ternary design space

(Figure 1b)

. We expect the shape and directional

tile connectivity

to influence the local deformation mechanisms accessible within the clusters, with [L] shaped

tiles showing bending

-

dominated local deformations and straight [

-

] tiles showing stretching

-

dominated behaviors.

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3

Figure

1

. Architecture of two

-

phase materials. (a) Selected isodensity tile geometries and

composite assembly. (b) Compositional design space in a ternary diagram. (A) and (B)

architectures are represented by red and blue circles, respectively. (c) Average c

oordination

<R> as a function of [T] tiles content. (A) and (B) reinforcing networks are represented by

red and blue circles, respectively. (d, e) Representative (A) and (B) architectures (d and e,

respectively). (f, g) Close

-

up view of meso

-

structures tha

t populate (A) and (B) architectures

in (d) and (e) respectively. Yellow, green, cyan, and blue represent 4, 6, 8, and 10+ tiles

meso

-

structures, respectively. (h) Meso

-

structures distribution in (A) (red bars) and (B)

architectures (blue bars). (i) Exampl

e of meso

-

structure with labeled coordination and

bridges. (j) Expanded version of (i). (k) Comparison of bridge length and their frequency for

(A) and (B) architectures (red and blue, respectively).

Network characterization

We evaluate the properties of the reinforcing networks using frameworks developed to

describe covalent random networks (Supplementary Discussion 2), at two hierarchical

scales. At the global scale, we evaluate the average coordination of the materials at c

onstant

density, and at the local scale, we analyze how growth rules affect the formation of

characteristic meso

-

structures. We evaluate the average coordination <R> in the reinforcing

networks, accounting for the presence of dangling bonds, unconnected l

igaments at the

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4

network edges (Figure 1c).

[28,29]

Scaling linearly with the volume fraction of 3

-

coordinated

tiles, we expect <R> to influence the global mechanical properties, like strength and

stiffness, as reported in other amorphous materials systems

.

[30

–

32]

To understand the effect

of the reinforcing network architecture on the composite properties, we compare two

different compositions with significantly different average coordination: (A)

-

networks (35 [T],

10 [

-

], 55 [L]), dominated by 2

-

coordina

ted tiles and floppy modes; and (B)

-

networks (80 [T],

10 [

-

], 10 [L]), dominated by 3

-

coordinated tiles and that are purely rigid (figure 1b and figure

1c, red and blue circles respectively).

Despite having the same reinforcing and matrix phase volume frac

tions, (A)

-

and (B)

-

network reinforced composites (NRCs) form different local meso

-

structures, defined as the

matrix domains enclosed by reinforcing network (Figure 1d and 1e). While the average

coordination of the reinforcing network explains the global m

echanical behavior of the

materials, studying the meso

-

structures that pattern each composite is key to understand

their local properties. First, the meso

-

structures are categorized and mapped based on size

and number of constitutive tiles (Figure 1f and 1

g). Then, their surface distribution is used to

indicate the texture of (A)

-

and (B)

-

NRCs (Figure 1h). Additionally, the number density of

each meso

-

structure (Figure SI 2), their angle of orientation (Figure SI 3), and the effect that

small meso

-

structure

s have on their surroundings (Figure SI 4) are important descriptors of

these architected composites.

We characterize the reinforcing networks by drawing parallels with the concept of network

bridges, often used in studying of the mechanical performance of

covalent random

networks.

[28,29]

A bridge (black solid lines, Figure 1i

-

1j) connects two 3

-

coordinated tiles,

considered anchored in the network (I

-

V white circles, Figure 1i

-

1j). It was demonstrated that

a bridge composed of 6 or more 2

-

coordinated til

es (red circles, Figure 1i

-

1j) forms a floppy

region within the network.

[28,29]

The presence of floppy domains in a stiff, yet deformable,

reinforcing network influences the local mechanical composite performance, resulting in a

globally more extensible

and deformable material (Figure SI 5). In this context, the presence

of an incompressible matrix phase is important to prevent large bridge deformations.

Because of the different content of 3

-

coordinated tiles, (A)

-

NRCs display a multi

-

modal

distribution o

f bridge lengths which are significantly longer than those of (B)

-

NRCs (Figure

1k).

Mechanical Properties

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5

Figure

2

. Mechanical characterization of composites.

(a, b) Engineering stress

-

strain curves

recorded during uniaxial ten

sion tests on plate geometries of (A)

-

NRCs and (B)

-

NRCs (red

solid lines in a, blue solid lines in b, respectively). The solid black lines in (a,b) represent the

response of samples photographed in (c, d), respectively. The solid gray lines in (a,b)

repres

ent the response of the same (A)

-

NRC and (B)

-

NRC samples, without the matrix

phase. Fracture events in the reinforcing phase of (A)

-

NRCs and (B)

-

NRCs are indicated by

red and blue arrows in a and b, respectively, and in the reinforcing networks by grey arr

ows

(see also Figure SI 6).

(c, d) Fracture evolution in representative specimens of (A)

-

NRC and

(B)

-

NRC, respectively. The circles indicate the locations within the specimens that display

the signs of voids growth (circles in c and d, frame II and insets

in c and d, frame II, bottom).

(e, f) Digital image correlation (DIC) maps of the representative specimens of (A)

-

NRC and

(B)

-

NRC recorded at 0.5% strain (e and f respectively).

The DIC maps refer to the areas of

specimens highlighted by (*) in frame I of

(c) and (d).

Although (A)

-

and (B)

-

NRCs have the same volume fraction of reinforcement and matrix

phases, the difference in average coordination and bridge length and different meso

-

structure populations influence the mechanical properties at both global

and local scales.

To

measure experimentally the mechanical properties of the chosen architectures, we additively

manufactured composite samples using a polyjet printer (Stratasys Objet500 Connex3).

Recent studies have focused on determining experimentally

the mechanical and physical

properties of objects printed by polyjet printing and shed light on the relationship between

the printing parameters and the final performance of the part.

[33

–

35]

In our study, a stiff

viscoelastic resin (VeroWhite Polyjet Resin) and a soft elastomeric resin (TangoBlack Polyjet

Resin) were chosen for the reinforcing phase and matrix phase, respectively. Both resins are

commercially available, and their constitutive

properties fall within ranges reported in

literature (Figure SI 7).

[18,36

–

38]

We combined these two materials in a polymer composite with

a volume fraction of reinforcing phase of 0.3. At this volume fraction, we observed that the

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composites display a d

esired tradeoff between rigidity and extensibility (Figure SI 8), while

the reinforcing network thickness is one order of magnitude larger than the polyjet printer

resolution limit (Figure SI 9). To characterize their mechanical response, we

performed plat

e

tension experiments and confirmed that at the global scale, the purely rigid

-

like (B)

-

networks

achieve higher strength and higher stiffness than the (A)

-

networks (Figure 2a and 2b).

Despite a significant difference in the global mechanical properties, th

e composites display

similarities in the local scale mechanisms that determine the initiation and propagation of

fractures.

Due to the remarkable adhesion properties between the two resins used in this

study

[39]

, fracture initiation does not occur at the

interface between the matrix and the

reinforcing network, in either pristine or pre

-

notched samples, but within the matrix (Figure SI

7).

Void nucleation in the matrix phase initiates the composite fracture process, similar to the

ductile fracture of meta

ls.

[40]

Void formation is followed by matrix detachment from the

reinforcing network, resulting in steady void growth (Figure 2c and 2d, I to III respectively). In

this propagation phase, the void growth and coalescence are hindered by the reinforcing

ne

twork bridges, which elongate as the sample undergoes tensile loading. Thus, the

average bridge length and extensibility before rupture become paramount, as these

characteristics predict the strain of the reinforcing network before failure (Supplementary

D

iscussion 3). After the sequential failure of the bridges (Figure 2a and 2b, red and blue

arrows respectively), we observe the complete loss of composite integrity.

The local composite architecture become key during failure, as strain localization in selec

ted

meso

-

structures leads to fracture nucleation and growth, as confirmed by 2D Digital Image

Correlation (DIC) at small strains (Figure 2e and 2f). Therefore, to design composites

capable of dissipating the most fracture energy, one must act on both the g

lobal and local

scale, tailoring the network rigidity and generating local meso

-

structures, to avoid localized

strain fields. To achieve this, we modify the connectivity rules of the growth algorithm.

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7

Figure 3

. Modified composites and their performance.

(a) Modifications of connectivity rules

and average coordination number as a function of [T] tiles (top and bottom, respectively). (b)

Engineering stress

-

strain diagram of Mod

-

(A)

-

NRCs (red solid lines) and of original (A)

-

NRCs (grey solid lines). The modu

lus of toughness (MOT) is reported for both composites at

the top of the diagram. (c, d) Fracture evolution in representative Mod

-

(A)

-

NRCs at 0.5% and

16% strain (c and d, respectively). (e) Engineering stress

-

strain diagram of Mod

-

(B)

-

NRCs

(blue solid lin

es) and of the original (B)

-

NRCs (grey solid lines). The MOT is reported for

both composites at the top of the diagram. (f, g) Fracture evolution in representative Mod

-

(B)

-

NRC at 0.5% and 16% strain (f and g, respectively). (h, i) Modification of microstru

cture

of (A)

-

and (B)

-

networks (h and i respectively) and measured polydispersity index (PDI) for

each network. (j, k) Variation in relative meso

-

structure orientation distribution of (A) and (B)

-

networks (j and k respectively). (l) Frequency of bridge len

gths for (A)

-

and Mod

-

(A)

-

networks (top and bottom, respectively). (m) Frequency of bridge lengths for (B)

-

and Mod

-

(B)

-

networks (top and bottom, respectively).

We changed the connectivity rules of a growth algorithm to increase energy dissipation

during

fracture in composites. By amending four tile adjacency rules (Figure 3a, top,

Supplementary Discussion 4, Figure SI 10), we prevented the formation of large floppy

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8

domains, which increased network rigidity, stiffness, and strength. The modified networks

displayed a purely rigid

-

like behavior, as shown by their higher average coordination than

the original networks (Figure 3a, bottom). We tested the effect of the modified reinforcing

networks on the composites‟ mechanical performance and fracture energy di

ssipation

through plate tension experiments. As a result of their higher coordination, Mod

-

(A)

-

NRCs

displayed higher ultimate tensile strength (UTS) and up to 60% increase in tensile stiffness

(Figure 3b red and grey solid lines respectively), while Mod

-

(B

)

-

NRCs had a 5% reduction in

stiffness as a result of the slightly lower average coordination (Figure 3e blue and grey solid

lines respectively). Although each composite begins failure at ~10% tensile strain, the

modified designs‟ damage tolerance dramatic

ally improved. At high tensile strain (up to

~16%), the Mod

-

NRCs carry a load of approximately 70

-

80% their UTS (Figure 3c

-

d and

Figure 3f

-

g). As a comparison, their original counterparts at the same tensile strain had

completely lost any load carrying cap

abilities, due to presence of sample

-

scale cracks and

coalesced voids, resulting from the extensive failure of the reinforcing phase.

Conventional

calculations of the stress intensity factor and local stress concentration field require making

assumptions b

ased on continuum mechanics: for composite materials, the reinforcing

feature sizes must be small compared to the size of the singularity zone, and the nonlinear

damage must be confined to a small region within the singularity zone.

[40]

In our irregular

composites these conditions are not satisfied: meso

-

structures sizes are in the order of

several mm (Figure SI 2) and crack nucleation occurs in multiple locations within the

microstructure (

Figures 2c, 2d, 3d and 3g

). In the present study,

to highlight ho

w these

simple modifications to the reinforcing networks influence significantly the energy dissipated

during fracture, we measured the modulus of toughness (MOT), taken as the area under the

stress

-

strain curve. Modifying the reinforcing networks in (A) a

nd (B) composites improved

the total dissipated energy during fracture of up to ~130% and ~60% respectively (Figure 3b

and 3e, top).

Considering global scale descriptors solely, like the average reinforcing network

coordination, is insufficient to explain

the higher strength of Mod

-

(B)

-

NRCs compared to (B)

-

NRCs. Thus, we evaluated the modified designs at the local scale, to investigate the effect

that simple modifications of the connectivity rules had on the meso

-

structures. First, we

notice by visual inspe

ction that the modified composites (Figure 3h and 3i, bottom) have a

significantly different internal structure than their original counterparts (Figure 3h and 3i, top).

The modified architectures feature a more homogeneous distribution of meso

-

structures,

which are quantified through the polydispersity index (PDI) (Figure 3h and 3i, Supplementary

Discussion 5).

The decrease in PDI by 33% for (A)

-

NRCs and by 20% for (B)

-

NRCs,

confirms that more stringent connectivity rules homogenize and coarsen the meso

-

st

ructures

sizes (Figure SI 11).

Furthermore, the modified composites feature meso

-

structures that

display a more homogeneous angle of orientation with respect to their original counterparts

(Figure 3j and 3k).

As a result of the more homogeneous size and o

rientation distribution of

domains, the modified composites are subject to a more homogeneous distribution of the

deformation during loading, preventing high strain localization (Figure SI 12) and

leading to

the multiple uniformly distributed void nucleati

on sites in the matrix (Figure 3d and Figure

3g).

Finally, we evaluated the effect of the modifications on bridges length distributions. In

Mod

-

(A)

-

NRCs, the increase in short bridges confirms that the newly generated networks are

more constrained and thus

rigid, compared to their original counterparts (Figure 3l).

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9

Conversely, Mod

-

(B)

-

networks have a distribution of bridge lengths that shifts towards larger

sizes and becomes multimodal, becoming like those of (A)

-

networks, suggesting the

generation of reinf

orcing networks with higher local extensibility and hence, higher bridging

capability (Figure 3m).

Figure 4

. Multi

-

architecture meso

-

scale assemblies. (a,b) Laminate assemblies: (A), Mod

-

(B), (A) and Mod

-

(B), (A), Mod

-

(B) (a and b respectively). The inse

ts highlight differences in

reinforcing architecture. Fracture evolution (I, II, III a and b respectively). (c, d) DIC maps at

0.5% strain in laminate assemblies. (e) Sketch of cross section of cortical bone (f) Cortical

bone inspired meso

-

scale assembly.

Mod

-

(A) constitutes osteon

-

inspired features (dashed

red semicircles), (A) constitutes the matrix phase. (g) DIC map at 7% strain and highlighting

strain distribution in cortical bone inspired assembly. (h,i) Fracture evolution at 11% and 21%

strain (h an

d i, respectively).

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10

We developed a method to control crack trajectory in network reinforced composites by

creating hierarchical microstructures that combine local rules, meso

-

scale assemblies and

macroscale connectivity networks at a constant density. We

drew inspiration from biological

composites like mother

-

of

-

pearl

[41

–

45]

and cortical bone

[46

–

48]

, which deflect incoming cracks

and dissipate fracture energy. Our meso

-

scale assemblies feature rational designs of 'strong

and tough' network portions combined with 'soft' network portions. We created two laminate

configurations with complementary meso

-

s

cale arrangements (Figure 4a I and Figure 4b I,

respectively) and found that the (A)

-

NRCs domains carry most of the strain regardless of

their spatial arrangement. For an applied 0.5% strain, (A)

-

NRCs domains are subject to

~0.8% strain whereas Mod

-

(B)

-

NRC

s domains experience as little as 0.3% strain (Figure 4c

and Figure 4d). We can thus control the fracture trajectory through domain assembly, since

fracture nucleates (Figure 4a II and Figure 4b II) and propagates (Figure 4a III and Figure 4b

III) in 'soft

' domains. These properties are also consistent with crack propagation observed in

single edge notch tension tests (SENT) (Figure SI 7). We take inspiration from the cross

section of cortical bone, composed of tightly packed osteons, enveloped by the cemen

t lines,

specifically designed to arrest and guide incoming cracks on tortuous trajectories (Figure

4e).

[49

–

51]

In our cortical bone

-

inspired assembly we embedded strong and tough osteon

-

inspired high coordination domains, in a floppy and low coordinati

on matrix domain (Figure

4f). At 7% strain, it is already visible how the strain localizes in the floppy portions of the

composite (Figure 4g), leading to fracture nucleation in the central matrix area (left side,

Figure 4h), that is then arrested as it ap

proaches the opposite osteon

-

domain (right side,

Figure 4h). Meanwhile, crack nucleation above and below the plane of propagation initiates

the desired process of re

-

nucleation and re

-

direction of the fracture, critical to deflect its

trajectory (red arrow

s, Figure 4h) and to successfully shield the osteon domains (Figure 4i).

Conclusions

In this study, we developed architected composite materials that exhibit a high degree of

hierarchical order through material design. By utilizing a virtual growth algori

thm, we

manipulated the local connectivity between isodensity tiles, resulting in the formation of

larger meso

-

structures, which were merged to create sample

-

sized assemblies with

predetermined spatial arrangements. This approach enabled tailoring the mech

anical and

fracture properties of the architected composites, at the local and global scales. We envision

that the use of different sets of starting tiles and the combination of different reinforcing

-

and

matrix phases, will allow to fine

-

tune the activati

on of desired reinforcement and fracture

energy dissipation mechanisms. Building on our proof

-

of

-

concept observations, we

hypothesize that controlling the spatial arrangement and continuity between the soft and

hard phases can be used to prevent interfacia

l failure, while their intentional design can

facilitate the precise spatial distribution of fractures in architected composites.

Supplementary Information

Supplementary Information is available from the Wiley Online Library or from the author.

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11

Acknowledgements

The authors thank P. Arakelian, K. Liu, T. Zhou, C. McMahan, and J. Boddapati for the

fruitful discussions. The authors acknowledge MURI ARO W911NF

-

21

-

S

-

0008 for the

financial support. T.M. acknowledges the Swiss National Science Foundatio

n for the

financial support.

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