# Toughening effects of out-of-crack-path architected zones

Julie Triclot · Thomas Corre · Anthony Gravouil · Véronique Lazarus

Received: 22 April 2024 / Accepted: 9 August 2024 / Published online: 21 August 2024
© The Author(s), under exclusive licence to Springer Nature B.V. 2024

**Abstract** The increasing use of architected materials has broadened the possibilities of mechanical behaviour. In this article, we aim to explore these new possibilities in terms of in-service behaviour, especially in terms of crack propagation by performing an in-depth study in the framework of Linear Elastic Fracture Mechanics (LEFM). The specific configuration studied here is the case where the architected zones are symmetrically positioned adjacent to the crack path and no propagation occurs within the zone. This problem is addressed both numerically and experimentally. Numerically, a path-following algorithm is used to simulate the crack propagation. Different toughening aspects of the addition of architected zones are identified. First, a temporary increase in crack propagation resistance is shown. It comes from a temporary increase of stored elastic energy in the architected zones, thus acting as mechanical springs. Second, a snap-back instability appears, linked to the release of the previously stored energy. It leads to a higher energy dissipated by the crack propagation process. Experi-

mentally, we evidence the possibility to reproduce the theoretical results using 3D printing. A good quantitative comparison is obtained between numerical and experimental approaches. This study shows that it is possible to improve crack propagation resistance while lightening the component by addition of architected zones outside the crack path. This opens up the way to tune finely, through the use of optimization tools, the crack propagation response.

**Keywords** Crack propagation · Architected material · Snap-back instability · Toughening mechanism · Path-following algorithm

## 1 Introduction

The rising of additive manufacturing methods has made it possible to consider a new class of materials for the design of structures: architected materials. These are characterised by a controlled configuration at the mesoscale, resulting in particular kinematics at the effective scale. They have generated a lot of interest as they exhibit unique novel properties such as negative effective Poisson ratio (Prawoto 2012) and can combine interesting properties such as relatively high stiffnesses with low densities (Ashby and Bréchet 2003). This makes this class of materials of particular interest to industry, especially in the transport sector, e.g. in aircrafts where architected materials are already used (Castanie et al. 2020).

<footnote>J. Triclot (✉) · A. Gravouil
INSA Lyon, CNRS, LaMCoS, UMR5259, 69621 Villeurbanne, France
e-mail: julie.triclot@upc.edu</footnote>

<footnote>T. Corre · V. Lazarus
IMSIA, ENSTA Paris, CNRS, EDF, CEA, Institut Polytechnique de Paris, 828 bd des Maréchaux, 91762 Palaiseau Cedex, France</footnote>

<footnote>T. Corre
Nantes Université, Ecole Centrale Nantes, CNRS, GeM, UMR 6183, 44000 Nantes, France</footnote>
The growing industrial interest in these materials raises the question of their in-service behaviour, notably when facing complex loadings. More precisely, studies regarding specific fracture behaviour have revealed a wide variety of interesting phenomena. In particular, a strong anisotropy in crack propagation sometimes appears (Réthoré et al. 2017; Nguyen et al. 2019; Corre and Lazarus 2021). Different toughening aspects of the fracture modes in architected materials have been highlighted. For example, crack path dependent effects, like tortuosity (Carlsson and Isaksen 2020; Hossain et al. 2014; Abid et al. 2019a; Zavattieri et al. 2007), crack trapping (Liu et al. 2020; Gao and Rice 1989; Lebihain et al. 2023), combinaisons of crack deflecting and crack arrest (Xiong et al. 2024; Brescakovic and Kolednik 2023), inclusion by-passing (Lebihain et al. 2020, 2021), pinning/fingering (Vasoya et al. 2016) or crack branching (Noselli et al. 2013) have been reported. In nacre-like structure, a phenomenon of crack bridging has also been observed (Abid et al. 2019b). Other effects that don't rely on the crack path have also been reported, like the toughening induced by elastic heterogeneities (Hutchinson 1987; Fratzl et al. 2007; Hossain et al. 2014; Brodnik et al. 2020), remote effects of heterogenity (Morano et al. 2020; Luo and Turner 2022; Xia et al. 2015) or specific failure modes due to buckling at the crack tip (Gibson et al. 1989; Fleck and Qiu 2007). On the other hand, the presence of cavities has been shown to reduce fatigue lifespan (Kumar et al. 2015).

The main objective of the paper is to make sense of what is possible in terms of toughening mechanisms in relation to architected materials while fitting into a pre-established configuration and theoretical framework. It focuses on a particular configuration that differs from what is traditionally studied. The architected material is introduced outside the crack path, and more precisely, outside the process zone. This allows setting free of material non-linear local effects and thus staying in the well-established framework of Linear Elastic Fracture Mechanics (LEFM). By analysing in depth, both theoretically and experimentally, the effects induced remotely by the presence of architected zones, this article provides a detailed discussion of certain toughening mechanisms linked to purely remote geometric effects. Such a study allows to discriminate which additional non linear material effects are involved in more complex cases. It also provides a simple way to optimize porous patterns for tougher and lighter structures.

<figure><img src="image_1.png" /><figcaption>**Fig. 1** Configuration studied in this work: out-of-crack-path architected zones with mode I loading and a straight line propagation</figcaption></figure>

To this end, we study the particular configuration of a crack propagating between symmetrically positioned architected zones, while enforcing the crack to remain straight. Doing so permits to focus on the remote effect of the architected zones, disregarding the attraction of the porosities on the crack path (Misseroni et al. 2015). Understanding this effect is of fundamental interest but also practical for instance for cracks constrained to propagate along a weaker plane.

The outline of this article is as follows. In Sect. 2, the studied configuration is specified along with some recalls the LEFM framework that will be applied in the following. In Sect. 3, the problem is studied numerically: a path-following algorithm with finite element computations is used to show that the addition of architected zones has a significant and complex effect. These results are discussed to reveal how the observed effects can be viewed as toughening. In Sect. 4, a set-up including 3D-printed samples and digital image correlation is presented. It shows that toughening revealed by the numerical study can be reproduced experimentally. Then, quantitative comparison between numerical and experimental approaches is presented in Sect. 5, and a general conclusion is given in Sect. 6.

## 2 Presentation of the studied problem

### 2.1 Crack propagation problem with architected zones

The studied configuration is illustrated in Fig. 1. We consider a crack propagating in a straight line between
two symmetrical architected zones. These zones are rectangle areas where porous patterns are introduced. The problem is assumed to be 2D, all force and energy terms will be defined by units of thickness. Bulk forces are neglected and the loading is a pure mode I traction that can be reduced to a single imposed displacement $u$ or its reaction force per unit thickness $P$. The material is considered as linear, homogeneous, elastic and isotropic with elasticity constants ($E, \nu$) and fracture toughness $K_{IC}$. The focus is made on the global response during the crack propagation. Therefore, the values of interest are $u, P$ and the crack length $a$. The crack propagation problem is studied in the framework of Linear Elastic Fracture Mechanics (LEFM), which means that the non-linear material behaviour including plasticity are assumed to be confined to the crack tip.

## 2.2 Linear elastic fracture mechanics

In this section, we recall the basic concepts of LEFM that are used in this article, with an energy point of view. The fracture phenomenon can be understood by writing down the first principle of thermodynamics (Lemaitre and Chaboche 1990). Here, heat generation is neglected. The principle thus states that the variation of the internal energy $E^{int}$ and of the kinetic energy $E^k$ is equal to the power of the external forces $P^{ext}$:

$$ \frac{dE^{int}}{dt} + \frac{dE^k}{dt} = P^{ext} \quad (1) $$

In the case of LEFM the internal energy is composed of the elastic strain energy stored in the solid $E^{el}$ and the energy dissipated by the crack propagation process $E^{diss}$. Any other dissipation is neglected. $E^{diss}$ results from the creation of new crack faces $da$ with the corresponding fracture energy $G_c$:

$$ \frac{dE^{diss}}{dt} = G_c \frac{da}{dt}. \quad (2) $$

The elastic energy, on the other hand, is defined as a function of the stress ($\sigma$) and strain ($\epsilon$) fields in the volume $\Omega$ and can be related to the traction vector ($f$) and displacements ($u$) at the border ($\partial\Omega$) by using the equilibrium equations. Assuming no bulk forces, it writes:

$$ E^{el} = \int_{\Omega} \frac{1}{2} \boldsymbol{\sigma} : \boldsymbol{\varepsilon} \, dV = \frac{1}{2} \int_{\partial\Omega} f \cdot u \, dS = \frac{1}{2} u P. \quad (3) $$

These loading parameters can also be used to describe the external forces work at a given time ($t$):

$$ E^{ext} = \int_{0}^{t} P^{ext} dt = \int_{0}^{u} P du. \quad (4) $$

By defining the strain energy release rate as $G = \partial(E^{ext} - E^{el})/\partial a$, the energy balance writes:

$$ (G - G_c) \frac{da}{dt} = \frac{dE^k}{dt}. \quad (5) $$

The irreversibility of the crack propagation process imposes $da/dt \ge 0$. By studying the sign of the variation of kinetic energy, it appears that only $G = G_c$ will lead to quasi-static crack growth $\frac{dE^k}{dt} = 0$. $G > G_c$ leads to a dynamic propagation, with $\frac{dE^k}{dt} > 0$.

The Irwin formula in mode I for plane stress applications ($G = K_I^2/E$) allows linking $G$ and $G_c$ respectively to the Stress Intensity Factor (SIF) $K_I$ and the toughness $K_{IC}$ so that the criterion for quasi-static crack propagation in mode I, can be written in terms of SIF:

$$ K_I = K_{IC}. \quad (6) $$

## 2.3 Geometry of the problem

The geometry chosen to illustrate the effect of architected zones on crack propagation is a Tapered Double Cantilever Beam (TDCB) specimen, the dimensions of which are given in Fig. 2a. All the dimensions in Fig. 2 are given relatively to the distance $W$, that will serve in the following as a length unit value to define dimensionless parameters. The TDCB specimen is loaded by pins inserted in holes. The lower pin's position is kept constant, and the top one is subjected to displacement at a constant velocity. The choice of this global geometry is based on the simplicity of the loading device, its stable<fnref n="1" /> configuration in the case without architected zone, and its noted ability to promote straight cracks when architected zones are added. The geometry is constant through the thickness.

Two architected zones are positioned symmetrically relatively to the crack path. The pattern chosen here to illustrate the effect of architected zones is composed of square holes. The geometry is given in Fig. 2b. The obtained medium has a porosity of 0.55. The Voigt bound (Voigt 1887) indicates that this architected material is more compliant than the constitutive material. The architected zone in the TDCB sample consists of six by six square holes.

<footnote n="1">¹ Here the term stable refers to the fact that the imposed displacement has to be increased for the crack to advance.</footnote>
<figure><img src="image_2.png" /><figcaption>**Fig. 2** Dimensions of the problem, given as value relative to the reference dimension W. In experiments, W will be taken as 40 mm. **a** TDCB sample, **b** Pattern for the architected zone</figcaption></figure>

# 3 Numerical study of the effect of architected zones

## 3.1 Numerical methods

In this section, the numerical method to simulate the crack propagation is presented. A first paragraph deals with the general description of the algorithm. Then, details on the model are given.

### 3.1.1 Path-following algorithm

The general structure of the numerical method relies on a path-following algorithm (Rastiello et al. 2022; Singh et al. 2016). The idea is to solve the problem by iterating on the crack length, and not on the loading which is the most intuitive thing to do. The main advantage is that finding the load at a fixed crack length for quasi-static crack propagation (6) is a linear problem while finding the crack length at a given loading is not. This allows decreasing the number of computations necessary to represent the whole fracture process. Moreover, since the iteration parameter does not explicitly depend on the time, this procedure is relevant to the detection of instabilities (Crisfield 1981).

More precisely, for each crack length (a), the elasticity problem is solved by the finite element method for a given material (E, ν), geometry (W) and loading parameter (u). Two dimensionless numbers representing the stiffness (S) and the SIF (k) are computed:

$$S = \frac{P}{Eu} \text{ and } k = \frac{K_I \sqrt{W}}{Eu} \qquad (7)$$

For a given sample shape, these two values depend only on a/W and the Poisson ratio, as can be demonstrated by a dimensional analysis using the linearity

of the problem (cf. Appendix A). Then, the loading parameters corresponding to the quasi-static propagation condition $K_I = K_{IC}$ can be easily evaluated:

$$u_c = \frac{K_{IC} \sqrt{W}}{Ek} \text{ and } P_c = Eu_c S. \qquad (8)$$

With this process, a parametric force-displacement curves can be built by iterating over the crack length a.

### 3.1.2 Finite element model

In practice, the commercial software Abaqus is used for solving the elastic problem and further to evaluate the stiffness and the SIF at a given crack length. The TDCB sample (Fig. 2a) is meshed, including the architected zone that is explicitly represented. More details on the mesh and convergence are given in Appendix B. The crack is explicitly included in the mesh with Barsoum elements at the crack tip to capture the stress singularity (Barsoum 1974). The opening of the crack is made possible by relaxing the symmetry boundary condition at the left of the crack. The SIF is evaluated by the well-established interaction integral, which is a domain integral, near the crack tip (Gosz et al. 1998). In line with our previous work (Triclot et al. 2023), the plane stress hypothesis is chosen. The chosen boundary condition on the pin is to impose the average displacement of the holes<fnref n="2" />.

<footnote n="2">² This is done via a reference point linked to the holes with the Abaqus option of *uniformly distributed-coupling*. This means that the solution is restrained to the one with uniform reaction force to the holes.</footnote>
## 3.2 Simulation results

In this section, the numerical results are presented. The loading parameters are given in dimensionless forms, so that the only material parameter affecting the results is the Poisson ratio. Here, a value of $v = 0.49$ is chosen to match with the material used experimentally, but we have checked that similar qualitative results can be obtained for other values.

### 3.2.1 Computed loading parameters

The first results provided by the path-following algorithm is the critical displacement and force corresponding to the quasi-static propagation condition for each crack length. These are given in dimensionless forms in Fig. 3. The grey area represents the part of the propagation where the crack tip is between the architected zones. The results for a bulk sample (i.e. without architected zones) are also shown for reference. This specimen undergoes a very classical behaviour as the displacement required to propagate the crack increases smoothly (Fig. 3a) while the corresponding force decreases (Fig. 3b). In Fig. 3, points are introduced (labelled A, B, C and D) to help guide the analysis.

The first thing to note from these curves is that the addition of the architected zones has a very significant effect on the propagation phase. It can be noticed that this influence goes way beyond the grey zone. Regarding the pin displacement required to propagate the crack (Fig. 3a), different phases can be observed. First, before point A, the required displacement to propagate the crack is approximatively the same for the architected sample and the bulk reference sample. Then, the curve for the architected case goes above the reference one. It reaches a local maximum in point B. Between B and C the displacement required to propagate the crack decreases with the crack length. Then it increases again, and the displacement of point B is overcome at point D. After point D, the curves representing the two samples (architected and bulk) get closer together until they are almost indistinguishable. Regarding the force required to propagate the crack (Fig. 3b), in the beginning, the force for the architected sample is lower than for the reference one. The two curves decrease at a similar pace until point A. Between points A and B, the force computed for the architected case decreases very slowly as a function of the crack length, almost reaching a

plateau. Finally, after point B, the forces for bulk and architected samples are almost identical.

### 3.2.2 Force-displacement curves

From the values displayed in Fig. 3, it is possible to obtain the equilibrium state variables as a function of the pin displacement. The obtained curves are shown in Fig. 4. It includes both the equilibrium phase before propagation and the quasi-static propagation. The first phase is a linear loading obtained by drawing a straight line between the point of zero loading and the point computed at the chosen initial crack length. In the case of Fig. 4, this value is set to $a_0 = 0.4W$. Regarding the propagation phase, the crack length in Fig. 4a is obtained via a simple switch of axis from Fig. 3a. The force-displacement curve (Fig. 4b) is obtained by combining the two curves in Fig. 3 and cancelling out the crack length.

The points A, B, C and D from Fig. 3 are reported in Fig. 4 and represent the different phases. In the linear phase, the stiffness is lower for the architected case. At the beginning of the propagation, before point A, the crack length is similar for the two samples as the force remains lower for the architected one. Between points A and B, the crack length at the same level of displacement loading becomes smaller for the architected case, and it evolves at a slower pace. In term of force, a plateau appears and the curve for the architected sample ends up going above the reference one. Between points B and D, the curves in both Figs. 4a and 4b go back. The displacement values in this phase lead to several solutions for the crack length and the force. This phenomenon is at the origin of a snap-back instability that will be explained and analysed in Sect. 3.3. After point D, both the crack length and the force get closer to the reference one.

### 3.2.3 Energy terms

Once the equilibrium loading parameters are computed, it is possible to calculate the energies that come into play in the crack propagation phenomenon. The stored elastic energy in the specimen and the total external work are obtained using Eqs. (3) and (4) and plotted in Fig. 5. The energies are plotted as a function of the crack length, so the mechanical state of the sample is compared at the same crack length, as opposed to the previous paragraph where the comparison was made at
<figure><img src="image_3.png" /><figcaption>Fig. 3 Numerical results for the quasi-static propagation in the bulk sample and the one with architected zones, plotted as a function of the dimensionless crack length. a Pin displacement, b Reaction force</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 4 Numerical results for the propagation in the bulk sample and the one with architected zones, plotted as a function of the dimensionless pin displacement. a Crack length, b Reaction force</figcaption></figure>

the same displacement loading value. We checked that the difference between these two energetic terms, i.e. the dissipated energy has an affine relation to the crack length, as awaited from $G = G_c$.

In these energetic curves, the effect of the architected zone is also observed. Regarding the stored elastic energy (Fig. 5a), while it decreases throughout the propagation in the bulk specimen, in the architected specimen the crack advance is accompanied by an increase in stored elastic energy for a part of the propagation (after point A until a bit before point B). After this point, it decreases quickly until meeting the curve of the bulk sample.

Regarding the work furnished to get to a given crack length (Fig. 5b), it increases smoothly for the reference case while a more complex behaviour is once again observed for the architected case. For low crack lengths, the external work required is smaller for the architected

samples. However, between point A and point B, it increases quickly, so it ends up going above the reference curve. A slight decrease is visible between points B and C. After point C it increases slowly to meet the reference curve.

### 3.3 Discussion of the numerical results

The problem presented in this article could be analyzed using different points of view. Here we choose to focus first on how an increasing displacement experiment would occur. Then, an analysis on how the effect of architected zones can be viewed as toughening phenomena is presented. Finally, an extension for other patterns is studied.
<figure><img src="image_5.png" /><figcaption>**Fig. 5** Energies calculated at different stages of the propagation for both the bulk sample and the architected sample. The energies are computed assuming quasi-static propagation. **a** Stored strain elastic energy, **b** External forces work</figcaption></figure>

### 3.3.1 Experiment with increasing imposed displacement

The overall behaviour for an increasing applied displacement can be discussed with reference to Fig. 4. For the TDCB setup used here, in the case of a bulk sample, the initial linear phase with no propagation ($K_I < K_{IC}$) is followed by a softening phase. The softening due to crack propagation is characterised by a decreasing force to apply to maintain quasi-static conditions while the displacement increases. When architected zones are present, the propagation behaviour is more complex and can be decomposed into four phases.

First, in the linear phase and before point A, for the same displacement, the architected sample is characterised by a lower force than the bulk case, but the same crack length. In this phase, the addition of the architected zone has an effect of the stiffness of the sample but not on the crack advance, the crack tip being far from the zones for them to impact the SIF.

Second, between points A and B, many phenomena appear. This phase corresponds roughly to the stage at which the crack tip is between the zones. An observed phenomenon that yields particular significance is the ability of the architected sample to store more elastic energy as the crack advances (Fig. 5a). This means that, in this phase, more energy from the external loading is required to propagate the crack than in the reference case. Indeed, this energy has to cover the dissipation of the crack propagation and the increase of stored elastic energy. This results in a higher slope of the curve representing the architected specimen than the one representing the bulk specimen in Fig. 5b. In an increased

displacement setting, it results in a higher displacement loading necessary to propagate the crack (Fig. 3a) and thus a delayed crack propagation (Fig. 4a). This delay in crack propagation also induces a delay in the force drop, explaining the force plateau observed in the force-displacement curve (Fig. 4b). The architected sample ends up showing a higher force than the bulk one at fixed displacement, even if the porous zone decreases the overall stiffness of the sample. The explanation for this behavior comes from the fact that for all cases, the stiffness decreases when the crack advances. The delay in crack propagation postpones this decrease, and it compensates for the initial lower stiffness.

Third, we turn our attention to the phase between points B and D. It was identified in Sect. 3.2.2 that the same *u*-value leads to different solutions for the crack length and the force. This non-unicity is actually the hallmark of a snap-back instability. To describe this phenomenon more precisely, the force-displacement curve for the architected specimen is plotted again in Fig. 6. The specific points B, C and D are reported here. The curves calculated with the condition $K_I = K_{IC}$ and plotted in Fig. 4 correspond to the quasi-static path. The fact that the curve has a backward turn at point B means that to maintain the quasi-static propagation condition, the displacement loading must be reduced until point C is reached. After point C, the loading can be increased again. In a fully increasing displacement setting, the sample will go directly from state B to D in a dynamic manner to release the excess of available energy since $G > G_C$. This instability can also be observed on the energetic curve (Fig. 5b), where the external force work is computed in quasi-
static propagation settings. The decrease of this energy between points B and C shows that the loading has to be decreased to maintain stability.

Finally, after point D, the architected zone becomes less and less mechanically loaded. This traduces in the difference between the architected sample and the reference one is getting smaller until it is undistinguishable in the curves in Figs. 4 and 5.

### 3.3.2 Toughening effects

In this paragraph, the previously observed effects of the addition of the architected zones are analysed to show how they can be considered as toughening phenomena.

- Temporary toughening

The first toughening effect relies on what happens between points A and B. In this phase, the architected zone is able to store more strain energy, delaying the crack propagation. This effect can be seen as a mechanical spring storing elastic energy to eventually release it. It delays the propagation in imposed displacement setting. In a more general setting, Fig. 5b shows that the addition of the architected zone increases the total amount of external energy necessary to reach a given crack length.

More insight on this effect could be given by drawing a parallel with the known effect of heterogeneity (Hutchinson 1987; Fratzl et al. 2007). In these works, the variation of Young modulus was shown to have a shielding effect. In the current configuration, the variation of stiffness is achieved not by varying the modulus in the crack tip region but by introducing architected zone away from the crack path. The result of the current work allows showing that the effect of the elastic heterogeneity can be induced remotely and that it can be uncoupled from any local effect (e.g. extra crack nucleations) or crack path dependent effects.

This toughening effect can be qualified as temporary because at the end of the propagation, the state of the architected sample, defined by its crack length, pin displacement, reaction force and energy terms is the same as the bulk sample. This can be seen in Figs. 3, 4 and 5 where all the curves meet for high values of a and u.

- Toughening by snap-back instability

The second toughening effect is due to the snap-back instability. To analyze its effect on the fracture process, the energy balance at the end of the propagation is made

<figure><img src="image_6.png" /><figcaption>Fig. 6 Snap-back instability: distinguishing the quasi-static path and the increasing displacement path, and highlighting the additional contribution</figcaption></figure>

for the two cases: quasi-static (q-s) and dynamic (dyn). In the quasi-static case, the kinetic energy is negligible so the total external work comes from the stored elastic energy and the Griffith dissipation:

$$E_{q-s}^{ext}(a) = E^{el}(a) + E^{diss}(a). \qquad (9)$$

At the end of the propagation, when $a = W$, there is no elastic energy left in the sample, consequently:

$$E_{q-s}^{ext}(a = W) = E^{diss}(a = W) = G_c(W - a_0). \quad (10)$$

It means that the energy required to break the specimen depends only on the material and the propagation distance of the crack. Therefore, provided that the initial crack length is the same, the bulk and the architected specimen require the same amount of external work force to break completely. This explains why the areas under the curves in Fig. 4b are the same. In the case of the dynamic path, on the other hand, the kinetic energy cannot be neglected. This term provides a supplementary contribution to the total balance:

$$ \begin{aligned} E_{dyn}^{ext}(a = W) &= E^{diss}(a = W) + E^k(a = W) \\ &= G_c(W - a_0) + E^k(a = W) \quad (11) \\ &> E_{q-s}^{ext}(a = W). \end{aligned} $$

This additional contribution is illustrated in Fig. 6. It represents a second toughening effect of the addition of the architected zone: the appearance of the snap-back instability results in an increase in the energy needed to break the specimen compared to the quasi-static path and thus to the bulk specimen. This effect is similar to what have been observed by Roux et al. (Roux et al. 2003) when comparing cases of weak and strong pinning in heterogeneous materials. They showed that in the case of strong pinning, a dynamic phase is present and lead to apparent strengthening.
### 3.3.3 Extension for other patterns

In the numerical study presented in this article, toughening effects of the addition of square holes architected zones have been identified. To extend the analysis, the simulations are run again for different patterns composing the architected zones. In addition to the square holes pattern, structures of hexagonal cells (with positive and negative angles) of the same porosity are tested. A structure with round holes and a smaller porosity is also added to the analysis. The results are displayed in Fig. 7.

The different patterns lead to different force-displacement curves, nevertheless, similar qualitative phenomena can be observed. Indeed, in all the architected specimen, there is a phase where the force drop is less steep than for the bulk sample. This is linked to the capacity of the sample to store more elastic energy while the crack advances. In all the presented samples, there comes a point where the force at same level of displacement becomes higher than for the reference case. The temporary toughening phenomenon is therefore not specific to the pattern chosen to present the previous results. In addition, a particular phenomenon occurs for the negative angle hexagonal pattern (yellow curve). There is a phase in which the force increases with the imposed displacement in the propagation phase. This is due to the fact that when a crack advances, the increase in elastic energy is greater than the energy used to propagate crack. This results in a hardening response in this phase.

The amount temporary toughening is, to first order, linked to a lower stiffness in the direction perpendicular to the crack. The use of architected zones is not the unique way of achieving this low stiffness. However, they can present complex behaviours like strong anisotropy and auxetism. These can be taken advantage of in future applications. For example, the strong anisotropy can be used to have the lower stiffness that leads to the toughening mechanism while having a fairely high stiffness in the transverse direction.

Considering the toughening by snap-back instability, it appears for the hexagonal patterns but not for the rounded holes one for which no backward turn is visible on Fig. 7. It allows concluding that the snap-back instability is not specific to the square holes pattern, but it cannot be generalized to all patterns either. The impact of the shape of the patterns is observed to be complex, and to involve many parameters. An exten-

<figure><img src="image_7.png" /><figcaption>**Fig. 7** Numerically computed force-displacement curves for different paterns in the architected zone</figcaption></figure>

sive study is however far above the scope of this paper and left for future work. The shape of the pattern influence not only the presence of the snap-back instability but also its extend. Indeed, the negative angle hexagonal pattern leads to a bigger jump than the squared holes one. It means the increased dissipation by the snap-back instability can be tuned by changing the pattern and eventually finding values higher than the one presented in Fig. 6.

## 4 Experimental study of the effet of out-of-crack-path architected zones

In this section, the idea is to reproduce experimentally the same physical problem.

### 4.1 Experimental methods

3D printing is used to manufacture the samples to easily build the architected zones. The Projet 3510 HD printer from the company 3DSystems was chosen because its microdroplet projection and UV polymerisation process allows producing globally homogeneous and isotropic behaviour. The material used in this study is the Visijet M3 Crystal, it is a photo-polymerized resin. We have shown in our previous work (Triclot et al. 2023) that this material can be modelled using LEFM. The material coefficients were then determined: the Young modulus is $E = 1.3$ GPa, the Poisson ratio $\nu = 0.49$ and the fracture energy $G_c = 1.15$ kJ/m$^2$.
Samples having a 5 mm thickness<fnref n="3" /> are made according to the dimensions presented in Fig. 2a with W = 40 mm. To obtain a stable initiation, a notch of depth 25 mm is added to the design and an initial crack is propagated by fatigue loading (see Appendix C for more details). The initial length as defined in Fig. 2a is 17 to 19 mm. In addition, to obtain a rectilinear crack path, a groove is introduced to weaken the specimen in the mid-plane and thus make is less sensitive to slight symmetry defaults. The grooves are carved on each face with an utility knife; the positions are shown in Fig. 8a. The obtained depths are 0.7 to 0.9 mm. For the rest of the study, we assume that this groove does not affect the global behaviour and therefore it is not included in the 2D numerical model.

The TDCB specimens are then tested with an Instron 3342 tensile machine at an imposed crosshead speed of 0.1 mm/min. Instrumentation for Digital Image Correlation (DIC) is also set up. This includes speckle patterns made with spray paint on the samples and monitoring the test with a Nikon D300s camera mounted with an AF-S Micro NIKKOR 60 mm lens. The obtained images are composed of pixels which physical size is approximately 20 µm.

DIC allows the calculation of the experimental displacement field. Here, the software *Ufreckles* developed by Réthoré (Réthoré 2018) is used. It relies on the search of displacement fields along finite elements shape functions (Besnard et al. 2006). The DIC tool is used to measure the crack tip position. The chosen method is the projection on Williams series and the cancellation of the first super-singular term. This method has been described in numerous works (Roux and Hild 2006; Hamam et al. 2007; Roux-Langlois et al. 2015). It is included in *Ufreckles* and some details are given in Appendix C. The shape of the projection zone has been edited for this analysis in order to include a sufficient number of points while excluding the architected zone where the measurement is not precise due to the free surfaces and where the Williams series do not apply *a priori*. The projection zone used in this study is shown in Fig. 8b. The DIC method with Williams series projection allows measuring the crack position at each instant a picture is taken. To complete the knowledge of the evolution of the crack length during the test, a measurement on the crack faces is done. It gives the crack length at the exact time the instability starts. Further details on this measure are given in Appendix C.

## 4.2 Experimental results

Similar to the numerical study, both the architected sample and the reference bulk sample are tested. For the latter case, no groove was introduced. The force-displacement curves as measured by the tensile machine and the crack length obtained by DIC analysis are respectively shown in Fig. 9a and 9b. The grey area represents the phase where the crack tip is between the architected zones as in Figs. 3-5. For readability purposes, it is not represented on the force-displacement curves. On these latter curves (Fig. 9a), it can be observed that the architected sample is less stiff than the reference one in the linear phase. Then, the force decreases at a much slower pace than for the bulk sample, displaying a sort of force plateau. There comes a point where the force at the same displacement loading is higher for the architected case. In the end, a sudden force drop is observed which is characteristic of the apparition of the instability. The dynamic phase goes until the force is 0, meaning until the whole sample is broken.

Considering the crack length (Fig. 9b), it has to be noticed that the initial crack length for the bulk specimen is about 1.5 mm smaller than the architected one. This difference comes from the initial fatigue loading. Once the propagation start, the crack advance at a slower pace for the architected sample than for the reference one. Quickly after the start of the propagation, the crack length at the same level of displacement loading becomes lower for the architected case.

## 4.3 Discussion of the experimental results

The aim of this discussion is to analyse the experimental results in light of the results of the numerical study (Sect. 3.3). The curves presented in Fig. 9 are comparable to those in Fig. 4, including a lower crack length for the architected sample, the force plateau and the presence of an instability.

To further analyse the effect of architected zones, the energies can be calculated. The stored elastic energy and the external forces work are obtained from the

<footnote n="3">³ We have shown in our previous work (Triclot et al. 2023) that samples with this thickness and global dimensions exhibit plane stress conditions.</footnote>
<figure><img src="image_8.png" /><figcaption>**Fig. 8** Experimental methods: illustration of the introduced groove (a) and of the specific DIC methods (b)</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>**Fig. 9** Experimental force-displacement curve and crack length evolution for both a bulk sample and an architected sample. The curve for the architected sample corresponds to the part before the instability. **a** Force-displacement curve, **b** Crack length</figcaption></figure>

experimental force-displacement curves using Equations (3) and (4) in the pre-instability phase. The results, shown in Fig. 10, are to be compared with Fig. 5. Regarding the elastic energy stored in the specimen, the specific behaviour identified in the numerical curves (Fig. 5) is observed in the experimental values: in the architected case, the propagation is accompanied by an increase in the stored elastic energy during a large part of the test. There comes a point where, for the same crack length, the stored elastic energy is higher for the architected specimen than for the reference case. In terms of external work, the curves are difficult to compare as this energy depends on the initial crack length. To set free of this dependency, we propose the following correction to the bulk sample value: subtract the

energy $G_c \Delta a_0$ used to bridge the gap in initial crack length $\Delta a_0$. The obtained values with this correction are given in Fig. 10a. The toughening effect identified in Sect. 3.3.2 is also found in the experimental curve: there is a range of crack length values where the total amount of external energy required to reach is higher for the architected case.

Regarding the toughening effect due to the snap-back instability, the external work at the end of the propagation has to be compared. This value is not directly accessible on Fig. 10a. At the last DIC frame, the external work for the architected sample and the corrected value for the bulk sample are similar. However, the bulk sample is not entirely broken at this point so this value will most likely continue to increase. The architected
<figure><img src="image_10.png" /><figcaption>**Fig. 10** Energies by unit thickness computed from experimental force-displacement curves for both a bulk sample and an architected sample. **a** Stored strain elastic energy, **b** External forces work</figcaption></figure>

sample, on the other hand, breaks suddenly shortly after the last point and thus the total energy furnished will not increase much. Therefore, the fact that the instability does not stop until the specimen is completely broken prevents us from observing the toughening effect due to the snap-back. It calls for additional studies to investigate whether the excess of energy absorbed by the snap-back can be obtained in practice by playing with geometrical effects like increasing the span width (W) to move the free edge away.

To finish on the experimental study, an aspect deserves some discussion: the presence of the groove. In order to reproduce the toughening effects from the numerical study, a groove had to be introduced to guide the crack path as it is very sensitive to the default of symmetry. This is due to a negative value of T-stress (Cotterell and Rice 1980). It could be argued that forcing the crack path to be straight by making it more fragile in that direction is counterproductive if we want to increase the resistance to crack propagation. However, the toughening effects found in this study overcome the fragility induced by the groove. This is shown by the experimental study (Figs. 9 and 10) where the reference sample has no groove while the architected sample has one. The comparison between these two experiments shows the toughening effects. These effects are thus present in spite of the presence of the groove. Future work could be conducted in order to have a straight crack path without groove. For instance, the global geometry and loading of the problem could be modified in order to maximize the T-stress.

To sum up on the experimental results, the temporary toughening effect has been found in all its aspects. The

instability starts as predicted by the numerical model but does not stop, which prevents the observation of the toughening effect corresponding to the snap-back instability. This can be due to the fact that the numerical model is based on a quasi-static model, not taking into account dynamic effects like wave reflections (Bouchbinder et al. 2010). Therefore, it is not surprising that this model is unable to represent the crack propagation behaviour in the dynamic phase.

# 5 Comparison between numerical and experimental results

In the previous section, the experimental results have been analysed. The comparison with the numerical model has been addressed qualitatively by searching the numerically identified toughening effects. In this section, the quantitative comparison between the numerical model and experimental results is made.

## 5.1 Comparison on one sample

First of all, a single test is compared with its numerical simulation. The numerical curves (Fig. 4) are modified to match the experimental settings. In particular, the $a_0$ value is adjusted to that measured experimentally and the curves are dimensioned using the values of E, W and $K_{IC}$. The Young modulus used is the one identified on the linear part of the reference samples $E = 1.28$ GPa. Particular care is taken with the $K_{IC}$-value. Indeed, due to the complexity of the printing process the toughness of the material varies from one
<figure><img src="image_11.png" /><figcaption>**Fig. 11** Comparison between numerical and experimental curves for the architected sample. **a** Force-displacement curve, **b** Crack length</figcaption></figure>

sample to another. To avoid discussing this dispersion, the toughness was re-evaluated for the specific sample used in this comparison. This was done by evaluating the fracture energy as the slope of the dissipated energy versus crack surface curve and deriving the toughness using the Irwin formula. The details of this procedure can be found in Appendix C. For the specific case presented here, the value used to calculate the numerical data is 1.02 kJ/m². Finally, both the numerical and experimental curves for the architected sample are displayed in Fig. 11.

The graphs in Fig. 11a represent the force-displacement curves, numerical and experimental. The beginning of the instability as observed experimentally is also displayed. It is defined as the last measurement point before the sudden force drop. In this figure, it can be seen that the correlation between the two approaches is excellent in the linear part as awaited. During the propagation, the two curves are a little more different but this difference can be considered as small in light of dispersion usually observed and the complexity of the phenomena involved. The onset of instability is at a displacement very close to that predicted by the numerical model but the force is a little lower.

The graphs in Fig. 11b represent the crack length as a function of the pin displacement, for both the numerical results and the experimental ones as identified with the DIC. The point at the onset of the instability is also displayed. As the time laps between two pictures for the DIC are too large (≈ 20 s) to measure accurate results at this precise instant, the measurement of the crack length on the crack faces is used for this data. On this figure, it can be observed that the correlation between the numerical results and the crack lengths measured by DIC is excellent. We can observe a jump

between the last DIC point and the one at the beginning of the instability that is measured on the crack faces. This could be due to the fact that these two measurements are based on a different definition of the crack tip. While the DIC locates the crack tip at the origin of the asymptotic fields, the measurement on the crack faces is blurred by a damaged zone ahead of the crack tip. Therefore, this second measurement is expected to give a higher crack length than the DIC. Using the crack faces measurement, the onset of the instability appears for a slightly longer crack than predicted by the numerical model.

## 5.2 Comparison on a parametric study

A parametric study on the geometry is presented to extend the scope the comparison. Here, the chosen parameter is the length of the architected zone $l_x$ as defined in Fig. 2a. It was observed that the snap-back instability occurs when the crack tip is close to the right edge of the architected zone. Therefore, increasing the zone's length is expected to delay the instability. Values of $l_x$ are given here in terms of number of square hole patterns that fit within the zone length. Experiments are conducted with $l_x$-values between 4 and 9 patterns with a total of 19 tested samples.

To compare the numerical and experimental approaches on this parametric study, the different physical values at the onset of the instability are considered. The force, displacement and crack length are plotted as a function of $l_x$ for both numerical and experimental results in Fig. 12. The identified linear tendencies are very similar for the three physical values considered. Indeed, the differences between the observed slopes do not exceed 11%. In terms of absolute values, the
<figure><img src="image_12.png" /><figcaption>**Fig. 12** Parametric study: influence of the zone's length on the displacement (a), force (b) and crack length (c), at the onset of the instability. Comparison of the numerically and experimentally identified tendencies</figcaption></figure>

observation is in agreement with the previous sections: the instability starts at the same displacement but at a lower force and higher crack length than predicted by the numerical model. Therefore, these gaps seem to be systematic. Possible explanations for the force gap are the effect of the groove, not included in the numerical model, or the effect of the boundary conditions (Triclot et al. 2023). Regarding the crack length, the gap may be due to a measurement issue. As explained in the previous section, the measurement on crack faces is based on a different definition of the crack tip that includes a damaged zone. The numerical simulation, on the other hand, does not consider a process zone. Therefore, the DIC measurement is more suited to compare with the simulations.

## 6 Conclusion

In this article, the possibility of using architected materials to improve the crack propagation resistance has been addressed through the study of a specific configuration of out-of-crack-path architected zones.

It appeared that the architected zones have different toughening effects during the propagation phase listed below:

1. A temporary toughening effect: the architected zones can act as mechanical springs that are able to store elastic energy, therefore delaying crack propagation. This traduces in different significant improvements:

(a) A slowing down of the propagation relatively to the displacement loading. This results in a

force plateau and thus higher applied force from a certain point onward.

(b) An increase in elastic energy which results in a higher external forces work necessary to propagate the crack.

2. A snap-back instability: The energy stored in the spring can sometimes be released dynamically. In this case, the total amount of energy required to break the specimen is increased; it is a permanent toughening effect of the architected zones. We showed numerically that this amount of energy can be increased by tunning the pattern.

While the temporary toughening has been retrieved experimentally with a great accuracy, the snap-back instability led to a dynamic phase until failure, while it is predicted to be finite by the numerical study. It stresses the importance of taking into account the dynamic effects into the numerical model to represent this phase accurately. It also calls for additional experiments to see if the numerical results could be reproduced with a longer sample.

Finally, this study highlights that while the use of architected material is particularly popular for its capacity to alleviate structures, it can also be used to improve mechanical performance. The toughening effects presented here open up the prospect of using architected materials to tune finely the crack propagation behaviour. Beyond the fundamental interest of this study, this result can be of practical significance for instance when a crack is enforced to propagate on a weaker plane. The numerical study showed a dependency to the pattern shape used in the architected zone,
thus it opens the way to optimization for practical purposes.

**Acknowledgements** We thank David Roucou, Nicolas Thurieau and Lachène Cherfa for their help with the experimental set-up, Pierre Gantier for the microscope images, Christine Bilon Lanfray, Jerème Cavoret and Nicolas Tardif from Pole S.mart Rhène-Alpes Ouest for their help with 3D printing. We acknowledge founding provided by the French Ministry of Higher Education, Research and Innovation and the support of Agence de l'Innovation de Défense (AID) via AID-2018 60 0071 00 470 75 01 (project 2019 - Porocrack) and via Centre Interdisciplinaire d'Etudes pour la Défense et la Sécurité - CIEDS (project 2022 - FracAddi).

For the purpose of Open Access, a CC-BY public copyright licence (https://creativecommons.org/licenses/by/4.0/) has been applied by the authors to the present document and will be applied to all subsequent versions up to the Author Accepted Manuscript arising from this submission.

## Declarations

**Conflict of interest** The authors declare that they have no conflict of interest.

# A Appendix: Dimensional analysis of linear elastic crack propagation problem

In this appendix, the full parametrization of the crack propagation problem is detailed through a dimensional analysis that allows defining functions that govern this problem with a reduced number of parameters.

## A.1 Non-dimensionalizing mechanical problem at a fixed crack length

First, the mechanical problem with a given crack length and loading has to be addressed and non-dimensionalized. In the linear elastic fracture mechanics framework, for pure mode I, the propagation condition relies on the Stress Intensity Factor (SIF) $K_I$. Thus, the relation between this value and the loading parameters has to be identified.

The first step of the non-dimensionalization process is to list the physical values that appear in the problem. Here the geometric values are: a the crack length, W the maximum crack length, related to the width of the sample. The other dimensions of the sample are fixed relatively to W for this study and hence are not included explicitly in the equations. The material values involved are the Young modulus E and the Poisson ratio $\nu$. The

loading parameter is the displacement u and the outputs are the force by unit thickness P and the stress intensity factor $K_I$.

Taking W as the characteristic length and E as the characteristic pressure, dimensionless numbers are built for all the physical values. The Vachy-Buckingham theorem (or Pi theorem) states that each output can be expressed as a function of the other dimensionless numbers:

$$ \frac{P}{EW} = f_1 \left( \frac{a}{W}, \nu, \frac{u}{W} \right), \qquad (12) $$

$$ \frac{K_I}{E\sqrt{W}} = f_2 \left( \frac{a}{W}, \nu, \frac{u}{W} \right). \qquad (13) $$

Taking into account that the SIF and the force evolve linearly with the displacement, the expressions can be re-written as:

$$ \frac{P}{Eu} = S \left( \frac{a}{W}, \nu \right), \qquad (14) $$

$$ \frac{K_I \sqrt{W}}{Eu} = k \left( \frac{a}{W}, \nu \right). \qquad (15) $$

The functions S and k are thus representative of the mechanical problem at fixed crack length. They respectively represent the stiffness of the sample and the SIF at a given displacement loading. From this point forward, the Poisson ratio is considered as fixed, so the two functions depend merely on the dimensionless crack length $a/W$.

## A.2 Propagation condition

To handle crack propagation, the propagation condition $K_I = K_{IC}$ is added into equations (14) and (15), to find the loading parameters at propagation:

$$ \begin{aligned} u_c &= \frac{K_{IC} \sqrt{W}}{E} \frac{1}{k(a/W)}, \\ P_c &= K_{IC} \sqrt{W} \frac{S(a/W)}{k(a/W)}. \end{aligned} \qquad (16) $$

The functions S and k also describe the propagation phenomenon. They do not depend on either E or $K_{IC}$. The dimensionless loading parameters that allow plotting propagation curves independent on those values are:

$$ \begin{aligned} \frac{u_c E}{K_{IC} \sqrt{W}} &= \frac{1}{k(a/W)}, \\ \frac{P_c}{K_{IC} \sqrt{W}} &= \frac{S(a/W)}{k(a/W)}. \end{aligned} \qquad (17) $$
<figure><img src="image_13.png" /><figcaption>**Fig. 13** Mesh used for this study. **a** Global mesh for the half specimen, **b** Zoom on the radial mesh at the crack tip</figcaption></figure>

## B Appendix: Mesh and convergence for the numerical model

For evaluating the stiffness and SIF function a mesh is generated for each tested value of the crack length. They are generated by a remeshing algorithm using the Abaqus python library. Because of the symmetry of the problem, only half of the geometry is modelled. The mesh is displayed in Fig. 13 and is composed of linear quadrilateral elements. A radial mesh is generated around the crack tip with Barsoum elements at the direct contact with the tip. For a specimen of size $W = 40$ mm, the element size varies between 1.5 mm and 0.08 mm. A refinement is performed on the zones of the boundary conditions, close to the crack tip and in the architected zone. The obtained mesh is composed of approximatively 20 000 elements.

The convergence of the model has been checked. The use of elements 10 times smaller leads to a difference of the stiffness of only 0.2 % and 0.1 % for the SIF.

## C Appendix: Details on experimental methods

### C.1 Fatigue loading

In order to start the test from a real crack and hence have a stable initiation, we perform a step of fatigue loading that initiates a pre-crack. Beforehand, a razor blade is slid along the printed notch to facilitate crack initiation. The fatigue step is conducted on an Instron Electro-Pulse tensile machine. The loading is a sinusoidal force of frequency 4 Hz. The force applied is 40 ± 20 N. This allows to have a 2 to 4 mm crack after the notch in 20000 to 40000 cycles.

### C.2 Crack length measurement

In this study, two types of crack length measurement are used: one based on Digital Image Correlation (DIC) and one based on the observation of crack faces. These two measurements are complementary as they give information at different instants of the propagation.

First, a few details on the DIC method are given. This method relies on the Williams series (Williams 1957) which can be written as:

$$ \mathbf{u}(r, \theta) = \sum_{i=1,II} \left( \sum_{n=n_{min}}^{-1} b_i^n \boldsymbol{\phi}_i^n(\theta) r^{n/2} + b_i^0 \boldsymbol{\phi}_i^0 + b_i^1 \boldsymbol{\phi}_i^1(\theta) \sqrt{r} + \sum_{n=2}^{n_{max}} b_i^n \boldsymbol{\phi}_i^n(\theta) r^{n/2} \right) \quad (18) $$

where $\boldsymbol{\phi}(\theta)$ are known dimensionless functions depending on Poisson ratio, and $b_i^n$ are coefficients that depends on the loading and the material characteristics. Using UFreckles (Réthoré 2018), these coefficients are identified by projecting the experimental displacement field to the solution, using $n_{min} = -3$ and $n_{max} = 7$ (Henninger et al. 2010; Mathieu et al. 2012). They are first identified at an initial approximation of the crack tip position. Then this position is iterated to cancel out the first supersingular term ($n = -1$) (Hamam et al. 2007).

Considering the measurement on the crack faces, the procedure is illustrated in Fig. 14. Hand-located points are put at the visible transition between the quasi-static crack face and the dynamic one. Then, a parabola is fitted through these points, and the crack length is determined from the mean value through the thickness of this parabola. The fact that we can see the transition between the two phases is due to a modification of the roughness of the surface. It can be observed in Fig. 15
<figure><img src="image_14.png" /><figcaption>Fig. 14 Illustration of the identification of crack lengths on pictures of the crack faces</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Fig. 15 Transition between the quasi-static crack faces (right) and the dynamic one (left). Pictures taken with a microscope HIROX MXB-5000REZ</figcaption></figure>

where we see microscopic traces of damage on the right part that are not present on the left part.

### C.3 Measurement of the fracture energy

The fracture properties of the material are defined in terms of the fracture energy that can be evaluated for each sample from the mere experimental force-displacement curve and crack length measurement. For this, the energy release rate G is evaluated following its definition:

$$G = \frac{\partial (E^{\text{ext}} - E^{\text{el}})}{\partial a} \Big|_{\text{at fixed loading}} \quad (19)$$

The term $E^{\text{ext}} - E^{\text{el}}$ is plotted as a function of the crack length in Fig. 16 for an architected sample. During the quasi-static phase, this term represents the dissipated energy $E^{\text{diss}}$. The thickness used for this computation is always the total thickness of the sample, even when a groove is present. This means that the toughness

<figure><img src="image_16.png" /><figcaption>Fig. 16 Identification of the fracture energy for an architected specimen. Dissipated energy as a function of the crack length</figcaption></figure>

computed is actually an equivalent toughness in this case, suitable for including it in a 2D model. During the propagation, a linear fit can be conducted on this data, which allows validating that the propagation occurs at a constant G. In particular the criterion $G = G_c$ leads to the measurement of $G_c$. This is done by identifying the slope on Fig. 16, here $G_c = 1.02 \text{ kJ/m}^2$.

This procedure is conducted for each sample, architected or bulk. The results for these two categories of samples both average at $1.15 \text{ kJ/m}^2$. It validates that the crack propagates following the same criterion $G = G_c$. On the other hand, the presence of the groove in architected samples is not taken into account in this procedure. Nevertheless, no significant difference due to the presence of this groove in the architected samples but not in the bulk samples is observed.

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