# Fracture metamaterials with on-demand crack paths enabled by bending

Lucie Domino  Institute of Physics, Universiteit van Amsterdam, Science Park
904, 1098 XH Amsterdam, The Netherlands  Université libre de Bruxelles (ULB),
Nonlinear Physical Chemistry Unit, CP 231, 1050 Bruxelles, Belgium  Mariam
Beaure d’Augères  Institute of Physics, Universiteit van Amsterdam, Science
Park 904, 1098 XH Amsterdam, The Netherlands  Jian Zhang  Department of
Mechanical Engineering, Eindhoven University of Technology, Den Dolech 2, 5612
AZ Eindhoven, The Netherlands  Faculty of Mechanical Engineering, Delft
University of Technology, Mekelweg 2, 2628 CD Delft, The Netherlands  Shahram
Janbaz  Institute of Physics, Universiteit van Amsterdam, Science Park 904,
1098 XH Amsterdam, The Netherlands  Alejandro M. Aragón  Faculty of Mechanical
Engineering, Delft University of Technology, Mekelweg 2, 2628 CD Delft, The
Netherlands  Corentin Coulais  Institute of Physics, Universiteit van
Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands

###### Abstract

In many scenarios—when we bite food or during a crash—fracture is inevitable.
Finding solutions to steer fracture to mitigate its impact or turn it into a
purposeful functionality, is therefore crucial. Strategies using composites,
changes in chemical composition or crystal orientation, have proven to be very
efficient, but the crack path control remains limited and has not been
achieved in load-bearing structures. Here, we introduce fracture metamaterials
consisting of slender elements whose bending enables large elastic deformation
as fracture propagates. This interplay between bending and fracture enables
tunable energy dissipation and the design of on-demand crack paths of
arbitrary complexity. To this end, we use topology optimisation to create unit
cells with anisotropic fracture energy, which we then tile up to realize
fracture metamaterials with uniform density that we 3D-print. The thin
ligaments that constitute the unit cells confer them a strikingly distinct
response in tension and shear, and we show that by controlling the orientation
and layout of the unit cells the sequential progress of the crack can be
controlled, making the fracture path arbitrarily tortuous. This tortuosity
increases the energy dissipation of the metamaterial without changing its
stiffness. Using bespoke arrangements of unit cells, metamaterials can have
on-demand fracture paths of arbitrary complexity. Our findings bring a new
perspective on inelastic deformations in mechanical metamaterials, with
potential applications in areas as diverse as the food industry, structural
design, and for shock and impact damping.

From carving flints to cracking eggs open, fracture is a phenomenon that has
always been put to good use by humankind. The traditional approach is to
choose precisely the points of application of the load to control how the
material breaks, as is done in thin films for example [1, 2]. An alternative
approach is to tailor the material instead. For instance, highly anisotropic
materials such as nacre [3, 4], wood [5, 6, 7], or cuttlefish bone [8] guide
cracks along one direction—they are extremely tough if one tries to break them
in the other. Inspired by natural materials, scientists have devised materials
with increased fracture toughness [9, 10, 11, 12]. One can then wonder whether
artificial metamaterials could be designed to achieve even finer control over
their fracture properties, such as cracks of arbitrary complexity (Fig. 1a) or
with tunable energy dissipation.

Figure 1: Controlling a fracture path with unit cells with giant fracture
toughness anisotropy. (a) In this article, we address the challenge of how to
achieve designer cracks of arbitrary complexity. (b) Naive example of a step-
shaped crack path made of holes in a homogeneous medium. (c) Unit cell with
giant anisotropic fracture toughness, $J_{1}/J_{2}=15$. (d) Fracture
metamaterial with a crack forming the letter “U”, “V” and “ A”, see also
Supplementary Movie S1. Figure 2: Larger anisotropy in the energy release rate
increases both the crack length and the crack path complexity. (a) Three of
the unit cells considered with anisotropy ratio $J_{1}/J_{2}=1.7$ (left),
$J_{1}/J_{2}=15$ (middle), and $J_{1}/J_{2}=45$ (right). (b) Normalised von
Mises stress $\sigma_{v}$ in the 4 central unit cells of a 10x10 sample under
uniaxial tension, from finite element numerical simulation (see Methods
section for details). (c) Crack paths obtained in uniform samples, with
horizontal (left) or vertical (right) unit cells, see also Supplementary Movie
S2. (d) Measured force-displacement curves for the horizontal (dashed line)
and vertical (solid line) samples.

Many promising approaches to steer cracks have been reported in the
literature, such as designing specific notches in thin films [13], patterning
a sheet with cuts [14] or designing complex adhesion with a substrate [15].
Using anisotropy is another common route to avoid mode I fracture in natural
and synthetic composites [16, 10, 17, 18], in particular the use of
microfibers with a selected and/or varying orientation [19, 18]. Yet in all
those examples the crack path is ill-controlled (Fig. 1b). On-demand crack
paths have been instead achieved using either local change of density [20],
stiffness [21], asymmetry [22, 23, 24], crystal orientation in thin films
[25], and curvature [26, 27]. Such materials however are not load-bearing,
which limits their potential applications. Alternative strategies consist in
tuning the rigidity of the material to switch from a brittle crack to a
diffuse mode of failure [28], or changing the connectivity of a network of
meso-structured tiles [29], but in both examples this comes at the cost of
predictability of the crack path.

Figure 3: Role of anisotropy on crack length and mechanical properties. Length
of the crack measured in numbers of broken unit cells (a), dissipated energy
(b) and stiffness (c) as a function of the fracture anisotropy ratio
$J_{1}/J2$ of the unit cell (that takes values 1.7 (green upward triangles),
4.5 (blue diamonds), 15 (gray circles), 45 (red downward triangles)), measured
in the 10 $\times$ 10 samples presented in figure 2c, made with either
horizontal cells (full symbols) or vertical cells (empty symbols). Error bars
are smaller than the symbols size, and the dotted lines are a guide for the
eye.

To circumvent these limitations, we propose a scale-free strategy to achieve
on-demand crack paths in metamaterials created from virtually any brittle base
material. Rather than optimising the whole material for a given crack path,
which is unattainable with current optimisation techniques, we create
metamaterials whose building block is designed to have specific fracture
properties, namely extreme anisotropy in the critical energy release rate
(Fig. 1c). These are tiled to make metamaterials that, while keeping the
volume of base material at a fixed ratio, break along on-demand crack paths
(Fig. 1d). We demonstrate the design strategy on 2D unit cells and several
cases of emblematic fracture paths. Our findings illustrate a new fracture
mechanism, where the least strong elements bend before breaking. We
demonstrate that it is the very presence of these large bending deformations
that enables the crack to align parallel to the loading direction and hence to
follow an arbitrarily complex path. Our findings establish the use of slender
elements within metamaterials as a distinctive avenue for the control of
fracture.

## Unit cells with anisotropic fracture

We start by designing unit cells with highly anisotropic fracture behaviour.
These are designed to exhibit very different critical energy release rate
depending on the loading direction [30, 31, & references therein]. In other
words, they are very weak if loaded in one direction but very tough if loaded
in the perpendicular direction (Fig. 1c). The unit cells are obtained by means
of topology optimisation: starting from a square computational domain, the
optimisation algorithm iteratively optimises the placement and removal of
material to maximize the anisotropy in energy release rates for two loading
directions [30] (see Materials and Methods). Notably, we also set a constraint
on the volume of material, for instance at 50%. Fracture anisotropy is
quantified by the ratio $J_{1}/J_{2}$, where $J_{i}$ denotes an aggregation of
energy release rates computed throughout the unit cell’s boundary for the
$i$th loading case. This quantity can be tuned, with values ranging from
$J_{1}/J_{2}=1$ to $J_{1}/J_{2}=45$. We then tile the square unit cells to
create a metamaterial with the required fracture properties. The question we
address in this paper is how to choose the unit cells to create given fracture
properties, such as a certain crack path like the one shown in Fig. 1d, see
also Supplementary Movie S1.

## Tortuous cracks in uniform metamaterials

Let us first consider uniform samples, that is to say samples made with the
same unit cell, oriented either horizontally or vertically. We use four
different types of unit cells, three of which are shown in Fig. 2a. We first
consider a 100-unit cells metamaterial ($10\times 10$ arrangement) and we
perform a numerical analysis for a uniaxial static mode-I loading, _i.e._
vertical direction in Fig. 2b, where only the 4 central cells are shown (see
Materials and Methods). The results of this simulation reveal that the stress
(or strain) field in front of a notch has the highest values in the smallest
ligaments, even if they are not in front of the notch. This means that for
orientations of the unit cells where the weak direction is parallel to that of
the loading, the crack would tend to avoid mode I fracture (perpendicular to
the loading direction) and preferentially propagate parallel to the loading
direction.

Figure 4: Anisotropy enhances designer cracks. (a) Combs with an amplitude
$\ell$ increasing from $\ell=3$ (left) to $\ell=15$ (right), for $n_{x}=10$,
$n_{y}=19$, using the unit cell with fracture anisotropy ratio
$J_{1}/J_{2}=15$. The arrow in figure iv denotes a horizontal shortcut taken
by the crack, see also Supplementary Movie S3. (b) Sketch of the design
principle used: $\ell$ is the number of unit cells between between two
horizontally-tough cells in adjacent columns, and $n_{x}$ and $n_{y}$ are the
number of unit cells in the specimen in the horizontal and vertical
directions, respectively. (c) Measured crack length as a function of the
amplitude $\ell$ of the programmed crack path, for unit cells with a fracture
anisotropy ratio of $1.7$ (green upward triangles), $4.5$ (blue diamonds),
$15$ (dark circles), and $45$ (red downward triangles)). The uncertainty of
the measured length is smaller than the size of the symbol. The dashed line
represent the prediction $\ell\times(n_{y}-1)+n_{x}$, which corresponds to the
length of the programmed crack. (d) Maximum crack amplitude $\ell_{\max}$
achievable as a function of the fracture anisotropy ratio of the unit cells.
The dashed line corresponds to the prediction $\ell_{\max}=J_{1}/J_{2}$.

We now perform experiments on samples composed of $10\times 10$ unit cells,
that we fabricate using 3D printing (see Materials and Methods). These samples
are then put under uniaxial tension in the vertical direction in a tensile
testing machine (see Materials and Methods and Supplementary Information for a
calibration of the constitutive material). We show in Fig. 2c specimens where
the unit cell’s toughest direction is horizontal (left images) or vertical
(right images). All the samples have a notch on the left to make sure the
crack always starts at the same location111One exception is the sample with
$J_{1}/J_{2}=15$ in the vertical orientation, which has been tested without
notch.. The fracture path for samples with a horizontal tough direction is
always straight and horizontal. This is the shortest path a crack can take in
this sample. For vertically tough samples, the fracture path becomes more and
more complex as fracture anisotropy increases. Indeed, the crack path is
straight for the lowest anisotropy (left column in Fig. 2c), but becomes
longer for higher anisotropy, see also Supplementary Movie S2. It actually
spans the entire sample for the highest value of fracture anisotropy (right
column of the figure). As expected, aligning the strong unit cell direction
with that of the loading makes the metamaterial tougher (Fig. 2d). This
confirms that the stress and strain at break are always higher for vertical
samples than for horizontal samples.

Those higher stresses at break coincide with longer crack and are directly
controlled by anisotropy. Both the length and widths of the crack increase
with the unit cell anisotropy (Fig. 3a). The crack length stays equal to the
sample width for horizontally-tough samples, but increases rapidly with the
fracture anisotropy for vertically tough samples. This is because the higher
the anisotropy of the unit cell, the more stresses are concentrated in the
vertical direction (Fig. 2b), i.e., the crack is more likely not to follow the
horizontal direction.

Figure 5: (a-d) Design principle for crack steering in horizontal and vertical
directions, as well as $90\text{\,}\mathrm{\SIUnitSymbolDegree}$ turns. The
unit cells used here have a fracture anisotropy ratio $J_{1}/J_{2}=15$. (e-h)
Image sequence showing the fracture of a $12\times 12$ sample with a pre-
programmed crack path in the shape of the letter “S”. Images are taken at
times $0\text{\,}\mathrm{s}$, $23\text{\,}\mathrm{s}$,
$36.9\text{\,}\mathrm{s}$, and $54.2\text{\,}\mathrm{s}$, which correspond to
a global strain of 0, 0.016, 0.025 and 0.038, respectively. The disks denote
the fracture locations in the unit cells and the colors of the disks encode
time (colorbar in (m)). See also Supplementary Movie S4. (i-j) Zoom-in on
figures (e) and (f), showing four unit cells located either side of the
horizontal part of the crack path. (k-l) Zoom-in on figure (e) and (g),
showing four vertical cells located either side of the central part of the
crack path. (m) Force-displacement curve corresponding to the experiment shown
in (e-h).

Intuitively, longer cracks release more fracture energy since they break more
molecular bonds and indeed, we see that the energy dissipated by the crack
increases with the fracture anisotropy ratio for vertically-tough samples, but
decreases for horizontally-tough samples (Fig. 3b). This is because we
maximize the fracture anisotropy ratio not only by making one direction
tougher, but also by making the other direction weaker. For the most
anisotropic sample, this leads to a 100-fold difference in the dissipated
energy between the vertically and horizontally-tough samples. Interestingly,
the stiffness of the samples barely varies for all samples, as shown in Fig.
3c, and is around $290\pm 77.\text{\,}\mathrm{MPa}$. We interpret this fact by
two compensating effects during the elastic loading phase. In both horizontal
and vertical orientations, the stress is concentrated in the ligaments
parallel to the loading direction. While those ligaments are thinner for the
horizontal samples, they are also shorter than the thicker ligaments of the
vertical samples. These findings suggest two important conclusions: First,
anisotropy in energy release rate can be used to steer cracks in heterogeneous
structures, and cracks tend to propagate along the weak direction of the unit
cells, _e.g._ , vertically in samples shown in Fig. 2c. Second, anisotropy
allows us to control the energy dissipated by the crack without compromising
the stiffness of the structure. Achieving tortuous fracture paths is a
particularly interesting approach for applications where both a specific
energy absorption and structural stiffness are required.

## Maximising crack length

Figure 6: On-demand fracture. (a-b) Design principle for crack steering in the
diagonal direction. (c) Specimens made with 10 $\times$ 10 unit cells after a
unidirectional tension in the vertical direction, that break following a path
that outlines the letters “M”, “E”, “T” and “ A”. Note that the letter “E” has
been rotated by 90 degrees. Unit cells are 12 mm wide.

In the preceding section we investigated unit cells with different fracture
anisotropy ratio. Here we investigate how different crack path designs can
influence the total fracture length to obtain a maximum value—with its
corresponding maximum dissipated energy. To that end, and inspired by the
tortuous geometry of the crack in Fig. 2c (rightmost column), we arrange the
unit cells in a metamaterial in order to create comb-like paths. Starting from
a configuration that contains only vertically tough cells ($n_{x}\times n_{y}$
cells), we replace some of them with horizontally-tough ones in a staggered
order (see schematic in Fig. 4b), thereby creating a weak path for the crack
to follow; these weak cells are then separated horizontally by a single unit
cell and vertically by $\ell$ unit cells. We perform experiments with an
increasing number of $\ell$ to obtain the longest comb-like crack path, as we
show in Fig. 4aand in Supplementary Movie S3.

From left to right, the figure shows increasing values of $\ell$, from
$\ell=3$ (figure labeled i) to $\ell=15$ (figure labeled vi). As apparent from
the figure, the crack follows the prescribed paths, and the vertical “teeth”
of combs are one unit cell wide. For higher comb amplitudes
($\ell>\ell_{max}=8$), the crack takes horizontal shortcuts (see, for
instance, the orange arrow in the figure labelled iv). This makes the fracture
path shorter than what we expect. We repeat the experiment for other unit
cells, and measure crack length (in number of broken unit cells) as a function
of $\ell$ (Fig. 4c), for all four unit cell designs (keeping the same colour
code as in previous figures). For low values of $\ell$, we see a linear
increase of the measured crack path with $\ell$ for all unit cells, apart for
the least anisotropic one (in green). This linear increase follows the
geometric law $\ell\times(n_{y}-1)+n_{x}$ (shown as a dashed line), which
simply confirms that the crack follows the target path. The experimental data
points depart from this prediction for values of $\ell$ larger than a certain
threshold $\ell_{\max}$, whose value depends on the fracture anisotropy ratio
of the unit cell. As expected, the larger this ratio, the larger is the value
of $\ell_{\max}$ at which the designed path is no longer followed; this value
is, respectively, equal to 1, 6, and 11, for the unit cells with an anisotropy
ratio 1.7, 4.5, and 15. For the most anisotropic unit cell (in red) the
samples always broke along the path, even for the largest sample we could
manufacture (with $\ell=33$).

Fig. 4d illustrates the maximum comb amplitude $\ell_{max}$ measured
experimentally as a function of the fracture anisotropy ratio of the unit
cell. For the most anisotropic unit cell, we could not measure $\ell_{max}$
(indicated in the figure with a vertical arrow). The linear trend observed in
the figure suggests a direct correlation between the anisotropy ratio and the
maximum comb amplitude obtained. Breaking a vertical structure of the target
path (delimited by three red cells in Fig. 4b) requires to break $\sim\ell$
weak unit cells, while taking a shortcut requires to break one unit cell in
the strong direction. For $\ell\gtrsim J_{1}/J_{2}$ it becomes energetically
favourable to take a shortcut, and break a horizontally strong unit cell
instead of several vertically weak cells. The dashed line in Fig. 4d
corresponds to $\ell_{max}=J_{1}/J_{2}$, which agrees reasonably well with our
experimental data.

## Fracture mechanism

But what mechanism could lead to fracture that is parallel to the loading
path? We now design samples that break following more complex paths. We
consider paths that span the entire width of the sample, that will break the
sample into two (top and bottom) parts. We outline the desired path on a
square grid (e.g., the red lines in Fig. 1a), and we then use the design
principle in Figs. 5a-d to steer the cracks: we align locally the weak
directions of the unit cells with the path. This allows us to steer the crack
either horizontally (Fig. 5a), vertically (Fig. 5b) or at an angle (Fig. 5cd).
To explain the mechanism behind crack deviation, we design a material that
breaks with the shape of a step function, and that we show in Fig. 5e. This
exemplifies a situation where one might need to deviate a crack away from a
critical area.

In Fig. 5f, we see that the crack first propagates horizontally along the
bottom left and then along the top right of the sample (highlighted in blue),
see also Supplementary Movie S4. There, mode I fracture predominates, as we
show in Fig. 5ij: The horizontal cells break at very low strains (locally we
estimate that $\varepsilon\simeq 0.03$ from Fig. 5i and 5j) and with very
little deformation of the whole specimen. This instance corresponds to the
first large drops in the force (for a global strain of about $0.01$ in Fig.
5m). After these two horizontal sections break (in blue in Fig. 5f), the
vertical central part is strongly loaded in mode II. This leads to the
shearing of the vertical cells, which can locally accommodate large strains
without failure ($\gamma\simeq 0.13$ from Fig. 5k and 5l). This corresponds to
the second large drop in load curve, for a global strain of about $0.023$ in
Fig. 5m). The last unit cells to break (highlighted in yellow in Fig. 5h) are
located at the very top right, and we impute this to the fact that the loading
is not perfectly uniform over the whole specimen. We attribute the strong
anisotropy in the fracture response to the slender nature of the beams that
compose the unit cells. The beams within the unit cell bend when the latter is
sheared, allowing for larger deformations that on tension.

Figure 7: Reverse fracture. (a) and (b) Failed attempts at a Z crack, using
unit cells with anisotropy in energy release rates of 15 as well as diagonal
cells (see Materials and Methods for details). In (a) the crack is initiated
at the right side of the specimen, in (b) it is initiated in the middle by
removing a unit cell there. (c) Z shaped crack obtained in a specimen made of
circular unit cells, horizontal unit cells with anisotropy 15 and diagonal
unit cells, see also Supplementary Movie S5.

In simple terms, the fracture mechanism at play here is the following: in the
metamaterial under load, the first slender elements to break are the ones
loaded under tension— _viz._ they are parallel to the loading direction. As a
result of this failure, the stress within the metamaterial redistributes.
Because of this stress redistribution, the slender elements that are
perpendicular to the loading direction will be loaded under shear. Since they
are slender, they can bend and thereby accommodate a large strain before
breaking and hence appear as ductile.

Eventually though, those slender elements loaded under shear will break. This
sequential failure enabled by the bending mechanics of slender element and
redistribution of stress is key to achieving cracks of arbitrary complexity.

## Designer cracks

In this example of a step-shaped fracture, we see that crack steering is
achieved through a transition in fracture modes, _i.e._ , from mode I to mode
II enabled by bending.

As a consequence, the parts of the crack path in tension (mode I) break before
the parts that are in shear (mode II), which ultimately yields the desired
crack path. This strategy works well in most cases, and the designs studied so
far can be used to set only horizontal and vertical cracks, and a combination
thereof. To improve the sharpness of our designs and to expand the range of
patterns we can break, we add one more cell that was designed so that, in the
same square computational domain, the fracture anisotropy is maximized at a
$45\text{\,}\mathrm{\SIUnitSymbolDegree}$ angle (see Materials and Methods).

With this new unit cell, shown in Fig. 6a-b, we can program different crack
paths following the same principle outlined above with the horizontally and
vertically tough unit cells. We used all these unit cells to design and
fabricate samples that outline the letters “U”, “V” and “A” (Fig. 1c) and also
those that make up the word “META” (Fig. 6c).

###  .1 Reverse crack

We now turn our attention to even more complex crack paths, with a fracture
that at some point reverses direction. We opted for a zigzag design,
coincidentally described by the letter “Z” (see Fig. 7a-i). To program this
crack, we first use the design principle in Figs. 5 and 6, where we start with
a specimen made almost exclusively of vertically strong unit cells, apart from
two horizontal lines and a diagonal line. We introduce a notch on the right of
the sample to initiate the crack there. Once under tension, we see that the
crack propagates to the left, as expected (Fig. 7a-ii); however, as soon as it
hits the diagonal cells, the crack does not follow our programmed path (Fig.
7a-iii). Ultimately, the crack front reaches the second line of horizontally-
tough cells and the sample is broken into two parts, without showing the
expected programmed shape (Fig. 7a-iv).

We attribute this effect to a very large stress intensity factor in mode I
once the fracture front reaches the diagonally-tough unit cells (Figs. 7a-ii),
because at this stage the crack is rather long. To preclude this effect, we
decide to nucleate the crack in the middle of the sample, creating a notch by
removing the central unit cell (see Fig. 7b); we also replace the horizontal
unit cells on the first and last columns with vertically tough cells, so that
the crack does not nucleate on the edges. Once under tension, we observe that
the first unit cells that break are the horizontally-tough ones located at
either side of the diagonal line, since these are easier to break. Next the
vertical cells in the central part break but not the diagonal cells (Fig.
7b-iii). Instead, the crack has taken a shortcut, and breaks the sample into
two, creating a shape that still does not resemble the letter “Z” (see Fig.
7b-iii).

The vertical cells located around the central part are too easy to break in
mode I, which explains why the crack tip takes a horizontal shortcut. We thus
decide to take our design one step further, and to replace the vertically
tough unit cells with those that have the lowest energy release rate when
subjected to biaxial loading: a square cell with a circular hole [30]. The
specimen obtained is shown in Fig. 7c-i) and it consists mostly of circular
unit cells with the same volume fraction of material as the topologically
optimised ones (i.e., 50%)—in addition to horizontally- and diagonally-tough
cells that delineate the letter “Z”. Once under tension, the horizontal cells
break first as before (see Fig. 7c-ii), but then the crack propagates to the
center (Fig. 7c-iii) and then diagonally as originally programmed. The sample
is finally broken into two halves with the intended shape. This path with
extreme turning angles is a powerful illustration of the potentiality of our
design principle for crack steering.

## Conclusion

We have shown that metamaterials based on topology optimised unit cells open
promising routes for fracture control. Their versatility stems from the
slender struts that compose the unit cells, making it break readily in tension
but bend significantly in shear. The shape of the crack can therefore be
programmed into the material, even for intricate paths.

Unlike other strategies that use complex microfabrication techniques or
complex loading sequences, we design a metamaterial whose fracture behaviour
is solely dictated by its geometry. This type of metamaterial is thus easy to
create once the unit cell designs are known, and we posit that this principle
generalizes to other base materials and other scales, as well as other loading
scenarios (e.g., compression, bending, twisting). The combination of bending
and stretching had been already used in the context of double-network
elastomers with enhanced toughness [33, 34] so a natural question would be to
ask whether cracks of arbitrary complexity could also be achieved in such
materials.

Further research is however required to extend our strategy to ductile
materials, as we used linear elastic fracture mechanics and brittle base
materials. Similarly, multi-phase metamaterials would be promising to study,
which would imply solving for solid-solid interfaces within the unit cell.
More importantly, while in 2D the crack design seems somewhat straightforward
(we can rely on intuition), the true potential of these design principles will
be unleashed when we explore to 3D fracture.

For these future lines of research we deem necessary to understand the
underlying mechanisms that governs the extreme fracture behaviours we observe:
Is the bending of the thin trusses essential for crack steering? Or could the
same level of anisotropy and crack steering capacity be obtained in simpler
unit cells design (_i.e._ without thick and thin trusses)? It will also be
interesting to assess how well these metamaterials perform in practical cases,
such as strong impacts rather than quasi-static loading. This will be the case
in most applications where fracture is unwanted, such as mechanical fuses and
impact mitigation, but also food or even everyday examples such as the
shattering of one’s phone screen.

(a)(b)(c) Figure 8: Computational domain used to design the unit cell for
shear loading: (a,b) Two shear loading cases in (2); and (c) Reduced
computational domain due to symmetry.

## Materials and methods

### Design of the unit cells

The systematic design of a unit cell with highly anisotropic fracture behavior
is conducted in this work by means of topology optimization (TO). In a
nutshell, TO is an iterative procedure that combines finite element anaylisis
(FEA) with a gradient-based optimizer. Specifically, at each iteration of the
optimization, FEA is conducted to evaluate the objective function—in our case
a quantification of fracture anisotropy—and sensitivity information is
extracted and used by the optimizer to improve the design for the next
iteration. The zeroth value of a level set function is used to describe the
topology, and the method of moving asymptotes (MMA) is used as the optimizer.

In our work we build some concepts of linear elastic fracture mechanics (LEFM)
into a topology optimization framework that uses an enriched finite element
formulation to evaluate the objective function and topological derivatives to
evaluate energy release rates throughout the boundary of the unit cell from a
single FEA. For a given load case $i$, for instance compressing the unit
horizontally, energy release values obtained are then aggregated into a single
value

$J_{i}=\frac{1}{N}\sum_{j=1}^{N}G_{j},$ (1)

where $G_{j}$ denotes the energy release rates that is evaluated at $N$
locations (enriched nodes) along the unit cell boundary.

Then the optimization problem is formally stated as

$\begin{split}\text{minimize }&\Phi=\omega J_{1}-(1-\omega)J_{2},\\\
\text{such that }&\quad\bm{K}_{1}\bm{U}_{1}=\bm{F}_{1},\\\
&\quad\bm{K}_{2}\bm{U}_{2}=\bm{F}_{2},\\\ &\quad V=\bar{V},\end{split}$ (2)

where $V$ is the volume of solid material that is constrained to a target
value $\bar{V}$ (for instance 50% of the computatinal design domain),
$\bm{K}_{i}\bm{U}_{i}=\bm{F}_{i}$ is the discrete system of equations
corresponding to the $i$th loading case (either vertical or horizontal),
$\bm{K}_{i}$ is the corresponding stiffness matrix, $\bm{F}_{i}$ is the force
vector, and $\bm{U}_{i}$ is the degree of freedom vector. In (2) $\omega$ is a
factor that is used to give priority to one loading case over the other, so
for $\omega=0.5$ we give the same priority to both loading scenarios.

The design of unit cells for vertical and horizontal loading has been
thoroughly discussed elsewhere [30]. Here we showcase the methodology for
optimizing the unit cells that maximize fracture anisotropy when subjected to
shear loading. Fig. 8 shows the loading cases in (2) (Figs. 8a-b) and also the
actual computational domain used during topology optimization (Figs 8c), which
is reduced by taking advantage of symmetry. The final design obtained after
topology optimization can be visualized in Fig. 6b, where it can be seen that
unit cells are tougher along $45\text{\,}\mathrm{\SIUnitSymbolDegree}$ and
weaker along $135\text{\,}\mathrm{\SIUnitSymbolDegree}$.

For more details on the procedure herein, the reader is referred to a thorough
description by Souto et al. [30] and Zhang et al. [31].

### Fabrication of the samples

The samples were 3D printed with a polyjet 3D printer (Connex 500, Stratasys),
which has a precision on the order of 200 $\mu$m. Taking this limit into
account, the unit cells are scaled up to a size 12 mm to ensure a satisfying
rendering of thinnest parts the unit cells. We used a brittle polymer (Vero,
Stratasys) and printed the samples with a thickness 3mm. The largest sample we
could print was 33 unit cells long, which was the maximum the print bed could
hold. The printer also induces anisotropy due to the fabrication process, so
we made sure to always print samples in the same direction (see also
Supplementary Information).

### Testing

The 3D printed samples were then fastened in custom-made clamps and then
placed in a testing machine (Instron 5985) equipped with a 300 kN load cell,
which enabled us to impose a tensile displacement with an accuracy of 0.01 mm
and to record the force with $0.5\%$ of the reading accuracy. The sample was
then put under tension at an imposed displacement rate of 0.1mm/s, and the
experiment was recorded using a USB camera (ac2040, Basler with a 50 mm lens)
at 25 FPS.

### Simulations in COMSOL

We use the 2D solid mechanics module from COMSOL. We model a linear elastic
material and we impose an infinitesimal displacement of $\varepsilon=0.008$ at
the edge of a 10$\times$10 specimen, with a very long notch that spans 5 unit
cells and that is a quarter of unit cell in thickness. This method does not
allow us to predict fracture propagation per se, which would actually be
rather challenging given that the crack needs to nucleate numerous times to
propagate. As a proxy, we identify the loci of maximum von Mises stress, which
corresponds to the direction in which the crack is most likely to propagate.
Note that in Fig. 2b we present the 4 central cells, located around the tip of
the notch.

###### Acknowledgements.

The authors would like to thank C. Ederveen Janssen, D. Giesen, S. Koot, M.
van Hecke and D. Ursem for their technical help. We acknowledge funding from
the European Research Council under grant agreement 852587 and the Netherlands
Organisation for Scientific Research under grant agreement NWO TTW 17883. All
the codes and data supporting this study are available on the public
repository https://doi.org/10.5281/zenodo.11103085.

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