Multiscale analysis of brittle failure in heterogeneous materials

Abstract This paper presents a versatile model-free approach for linking the damage in highly heterogeneous materials at multiple scales. The proposed scheme evolves from phase-field modelling at the microscopic scale to simulate brittle failure, towards the estimation of the effective elastic, toughness and strength properties of the material at the mesoscopic scale, via a model-free coarse-graining technique. On one side, it’s shown that, in comparison with the classical homogenisation approaches, the considered upscaling method: (i) requires no RVE (representative volume element), (ii) can be applied when the statistical homogeneity of the material ceases to exist and (iii) when sharp localisations are present. On the other, (iv) the quasi-brittle behaviour of the material is justified without any assumption on the model at the mesoscopic scale. Most prominently this paper shows that (v) the consideration of an effective homogeneous continuum to substitute a microscopically heterogeneous one is dictated by the use of a much larger regularisation parameter than what has been classically established.


Introduction
The desire for better-performing materials has long been established, and expansion of the boundaries of the material property space has already been achieved in multiple ways, i.e., whether by manipulating the chemistry, through developing new alloys and polymers, or by manipulating the microstructure through thermomechanical processing [1]. Innovative materials for aerospace and automotive applications are recently requiring the improvement of the mechanical properties while reducing the structure's weight.
Accordingly, engineers have shown interest in controlling the architecture of the materials through thoughtfully designing them in a certain fashion, to acquire improved mechanical properties over their constituents. However, the use of those highly heterogeneous materials is bridged by some limitations, de facto, treating the heterogeneities for accurately simulating the complex behaviour of such structures requires huge computational resources. Thus, of course, it's appealing to describe a simpler nature of those materials.
A variety of upscaling methods were proposed to reveal the relations between the microstructural heterogeneities from one side and the behaviour at higher scales from the other [2]. The underlying principle of existing classical homogenisation techniques lies on the description of a structure with the help of a much smaller specimen, known as the representative volume element -RVE. This implicitly assumes the presence of two separated scales: (i) the microscopic scale that is small enough to capture the heterogeneities in the material, and (ii) the overall scale of the structure where the effects of the heterogeneities are expected to be smeared out, and on which effective material properties are considered [3]. The classical (first-order) homogenisation method is based on the construction of a boundary value problem on the RVE that allows the determination of the effective material properties at the higher scales [2]. In this case, the RVE should be big enough to statistically capture the heterogeneities and be constitutively valid, yet small enough to be considered as a volume element of continuum mechanics. More details on the classical homogenisation method can be found in the literature, and the respective work outlines the following limitations: upscaled deformation modes of an RVE for a first-order homogenisation are linear; the first-order methods cannot take into account the size effects, nor large gradients of deformation, nor localisation, i.e., also, in case of large gradients, even materials with small microstructure cannot be accurately modelled [4]. Moreover, the first-order schemes do not work for softening materials. To surpass these problems, an extension to higher-order approaches has also been addressed [4,5]. The solution of the microscopic boundary-value problem in the case of higher-order computational homogenisation is effortless, yet allows for an enriched upscaled continuum with higher-order strain and stress fields [4].
Although the higher-order techniques are able to treat softening materials, they present their limitations: localisation bands beyond a quadratic nature for the displacements cannot be resolved, i.e., softening materials in the presence of sharp localisation regions from the presence of a crack and/or high heterogeneities [6].
Aside from determining the effective material properties, and with the increasing demand on architectured (highly heterogeneous) materials, there's a relevant need for incorporating small-scale mechanisms of deformation and damage to essentially assess reliability and lifetime of structures within reasonable computations. As damage localises in narrow regions in a considered continuum, the length scale that determines the variation of the defect falls below the considered scale of the mechanical fields (RVE) leading thus to what is known as gradient effects [7]. Gradient theories emerging from the multiscale nature of the mechanical framework are based on the enrichment of the classical continuum description with additional terms; those allow taking those gradient effects into account [8]. When the constitutive equations at the higher scales are difficult to write, general methods based on concurrent finite element simulations (FE2) can be applied [9]. FE2 methods do not require any constitutive equations because all non-linearities come directly from the homogenisation of microscopic quantities after applying localisation rules to determine local solutions. Interests are presently concentrating on the development of a continuous-discontinuous homogenisation scheme, to allow the assessment of the presence of both micro and macro cracks, and where localisation bands are incorporated at the macroscale [10].
Recently, work has been done on deriving a homogenised cohesive law at the macroscale from computations of crack propagation in a microscopic sample [11]. In [12], X-FEM approaches are used to incorporate the discontinuity at the macroscale, but as previous methods, this technique relies heavily on the principle of separation of scales as well as the presence of an RVE and it remains a homogenisation technique where a small part of the domain is considered to extract the full response of the structure; plus, both those methods rely on concurrent multilevel finite element (FE2) which is computationally expensive, and requires the difficult task of writing a consistent homogenisation scheme to link the scales [9]. Moreover, the need for enrichment of the description of the multiscale problem is directly perceived [13,14,15]. Although those methods are theoretically prominent, their experimental applicability remains questionable. In [16], a different approach has been proposed, where the effective toughness of the heterogeneous media was directly evaluated a priori (without concurrent computations). Recently, [17] followed the work of [16] to identify the different parameters of a damage model at the mesoscale by fitting a typical force-displacement response on a heterogeneous structure. Yet, as the effective material properties are determined macroscopically from force-displacement responses, it's believed that micro-cracks and their influence on the structural responses fail to be taken into consideration.
The above mentioned methods stand as long as the separation of scales is prominent, or as long as an RVE can be well-defined, which naturally leads to a homogeneous description of the microstructures at the macroscopic scale.
Nonetheless, when the microstructure's heterogeneities and/or the damage distribution are not statistically homogeneous, the effective description is expected to be dependent on the position in space and becomes influenced by simultaneous interactions between the damage and the microstructure.  based on the actual physics in question at the scale of the heterogeneities.
Without any a priori on the material's behaviour, the herein proposed scheme provides a genuine evaluation of the effective material and failure properties at the considered scales. The different scales of interest are presented in Section 2.1. The paper is organised as follows: the computational method considered for the simulations at the microscopic scales is briefly introduced in Section 2.2 and the proposed coarse-graining method is provided in Section 2.3. Application of the scheme for the analysis of mesoscopic damage on typical microstructures is then constructed and discussed in Section 3; the effective material and failure properties are assessed in a lower-cost and more straightforward manner.

General Statements
Let us briefly recall the multiple scales of interest at which damage problems can be tackled: the microscopic level is the level at which the material's architecture is prominent. In this study, reproducible lattices of pores are considered, e.g., Figure 1(a). Continuum mechanics apply; brittle failure occurs and can be simulated by the linear elastic fracture mechanics or its approximations (Phase-field Modelling [19], Eigen erosion [20]...). At the macroscale, the material is seen as a homogeneous bulk (Figure 1(c) [23] approach was developed. In the TLS approach, the damage zone is separated from the undamaged zone and the damage variable is linked to the level set function which itself is a parameter of the model. But with fewer assumptions on the model, the phase-field framework is believed to be more convenient for our study.
In this section, the phase-field approach as implemented in [24] is briefly presented; the choice regarding the method's parameters is based on the study led in [25].
Assuming small strains, the phase-field introduces the following energy functional for a cracked body in a regularized framework: Where u and Γ are the variables representing the displacement and the crack surface respectively. The crack surface being a function of a continuous damage variable α. α describes the material damage state: it takes the value 0 in the intact region of the material and 0 < α ≤ 1 to represent the crack.
E(u, Γ(α)) is the strain energy stored in the cracked body, E s (Γ(α)) is the energy required to create the crack according to Griffith Criterion -known as the fracture energy.
The sharp crack is smeared-out by a regularisation parameter l c [26], and the fracture energy is thus written as a function of the regularised crack density function γ(α, ∇α).
It's recalled that this approach uses the continuous field α to describe the discontinuities coming from the presence of a crack by the means of a crack density function γ. The term W u (ε(u, α)) represents the strain energy density in the cracked body, ε is the displacement symmetric gradient and g c is the fracture toughness. The regularisation parameter l c choice has also been previously assessed ( [27,28]). And it was shown that this phase-field modelling approach converges to the classical brittle failure when the regularisation parameter l c approaches 0. More details on the choice of the parameters considered for the finite element implementation are given in Section 3.1

Coarse-graining
When seeking to find continuum mechanics for the fracture process, out of information gathered at the microscopic scale, the challenge is to find an appropriate technique that takes into consideration the real solicitation of the material as well as the singularities coming from crack propagation and the presence of the heterogeneities. The micromechanical fields coming from phase-field simulations are upscaled by adapting the method from [29] and [18]: a physically consistent upscaling coarse-graining method that allows going from discrete probability density into an upscaled continuum ( Figure   2).
The parameters of this method are: • the convolution or the coarse-graining function φ that can be any sufficiently regular function with a local support: a variety of forms were studied and similar results were obtained; in this paper, the normalised Gaussian distribution (Figure 2(b)) of zero mean µ and a standard deviation σ is considered. It takes the following form: • the width of the convolution function l CG = w/2. In 3, w = 2 × 3σ: it's the most important parameter that defines the different length-scales at which the problem is inspected. The normalised Gaussian function can be rewritten as follows: • the discretization H considered for the coarser mesh (the support for In [18], a system of particles indexed by e is considered (Figure 2(a)), with known masses m e (t) and centres of masses r e (t) at time t. The coarse-grained mass density at position r and time t is given by:  Unlike in [18], continuum data is considered at the fine-scale from the micromechanical simulations. Let Ω 0 be the domain of interest in the microstructure, a discretization of Ω 0 into finite elements serves as a support for the coarsegraining computations. The coarse-grained mass density R(x, t) at position x in Ω 0 , at time t, is defined as the convolution between the microscopic density function ρ and the predefined coarse-graining function φ: For the sake of simplicity, the following notation is considered to replace the convolution: and the coarse-grained mass density R(x, t), at position x and time t, would be as follows: From this spatial/temporal definition of the coarse-grained mass density, and by imposing the mechanical conservation laws at both the microscopic and coarse-grained scale, expressions for the impulsions, velocities, displacements and stresses are obtained at different positions x and times t, at the coarsegrained scale. We start by recalling the conservation laws written at the microscopic scale; i and j denote the different directions in the considered space: • Balance of Mass

Balance of mass
A simple manipulation of (9) allows the computation of the expression for the velocity at the coarse-grained scale. Computing the convolution of both sides of the equation, one can obtain: The left side of the equation denotes the time derivative of the coarse-grained density R(x, t): Using the basic rule of the derivation of convolution, one can write: 13 Writing the balance of mass at the coarse-grained scale, with RV i denoting the impulsion P i at the coarse-grained scale: and by identification between (2.3.1) and (2.3.1), we can conclude that Identifying the coarse-grained impulsion, P i = RV i , and the microscopic impulsion, p i = ρv i , one can see that the coarse-grained impulsion is equal to the coarse-graining of the microscopic impulsion, which is not the case for the velocity field. The velocity at the coarse-grained scale is the ratio between the upscaled impulsion and the coarse-graining mass density: In this study, continuum mechanics is assumed to hold in all length scales involved; the derivation is restricted to small displacement gradients and the discussion is confined to two-dimensional quasi-static problems on perfectly solid materials. Therefore, the coarse-grained displacement U i and velocity V i fields have similar expressions, from (2.3.1): Next, it is natural to proceed with a strain calculation based on the coarsegrained displacements:

Balance of linear momentum
At the microscopic scale, the balance of linear momentum states: at the mesoscopic scale, a similar expression is expected with coarse-grained mechanical fields, to be written as: From the time derivative of the coarse-grained impulsion P i = RV i = ρv i φ (2.3.1), and using the basic rule of derivation, the expression of the stresses at the coarse-grained scale is determined: from the balance of momentum at the microscopic scale (10), we can write (2.3.2) as: It's here interesting to introduce what is called 'fluctuating velocity This velocity does not add any impulsion to the system, and the coarse-grained fluctuation impulsion vanishes as: Once, v i is replaced in (2.3.2) by v i + V i , the following equation can be written: Now writing the coarse-grained balance of linear momentum as a function of the coarse-grained variables, and by identification between (2.3.2) and (2.3.2) the expression of the stress at the coarse-grained scale is obtained: In quasi-statics, the dynamic terms v i v j will be neglected and the stress at the coarse-grained stress field S scale is found to be equivalent to the convolution of the microscopic stress field with the coarse-graining function: S = σ φ .

Application on Mesoscale Analysis of damage
In this section, we present detailed information about an application of the proposed scheme where we present the typical microstructures of interest and their corresponding micromechanical simulation followed by an in-depth investigation of the results.

Typical Microstructures
Due to their wide presence and uses, periodic and quasi-periodic microstructures are considered in the study, with the idea that any material would behave between a perfectly periodic one and a quasi-periodic material presenting long-range heterogeneities. For this purpose, hexagonal and kite&dart Penrose paving are studied. The materials' symmetry order (6 and 5-fold symmetry respectively) should lead to elastic isotropic equivalent .

Typical microstructures properties
Geometry Characteristic length(s) (µm) Hole radius (µm) Periodic 3000 750 Quasi-Periodic -Type 1 2270 and 3670 750 Quasi-Periodic -Type 2 2270 and 3670 750   for the Penrose kite&dart paving, there are two characteristic lengths. As mentioned previously, the microstructure is modelled by the drilled holes inside a bulk material which mechanical properties are presented in Table   2. ρ is the material density, E and ν are respectively the Young Modulus and Poisson ratio, l c is the internal length of the phase-field model and g c the fracture toughness. In order to meticulously compare the microstructures, the same hole radii r h = 750µm are taken, and the mean distances between the holes is fixed to d = 3000µm corresponding to a volume fraction of 78% for the periodic microstructure and 75% for the quasi-periodic ones.
The generated microstructures are shown in Figure 3, and their respective geometrical aspects are displayed in Table 1. Two types of quasi-periodic microstructures are considered, both based on the kite&dart Penrose paving.  one presenting specific quasi-periodic patterns. For the micro-mechanical simulations, the microstructure is put at the core of a Tapered Double Cantilevered Beam (TDCB) fracture geometry to provide crack growth stability from the tapered profile of the specimen [31,32]; the microstructure is surrounded by a homogeneous bulk material, the dimensions are put forth in Figure 4. Displacement boundary conditions are applied for the phase-field micro-mechanical simulation. In fact, the stability in crack growth provided by the TDCB specimen is believed to match the stability provided by the application of a surfing boundary condition [16]. In both cases, the crack evolves as it pleases inside the microstructure. In this study, the former -more straightforward -approach is adopted. We recall that we confine ourselves to the quasi-static evolution of cracks neglecting thus dynamics effects. The numerical discretization h = 200µm is thoughtfully adapted to the heterogeneities sizes and placements inside the domain and based on [25], the internal parameter of the phase-field model l c is set to 400µm.
Both lengths are much smaller than the structure's heterogeneities leading to mesh-independent crack initiation and propagation.
The influence of the microstructures on the crack propagation is prominent and put forth in Figure 3. For the periodic material, a simple linear crack path is obtained suggesting thus the presence of weak planes [30]. While Exploiting the adapted coarse-graining method, one is able to investigate the microstructure at different transitional scales by building density, displacement, strain and stress fields at each scale.

Density
As established in the method, the density field R(x) is first computed.
Effective density fields of the quasi-periodic type 1 microstructure at different coarse-graining scales l CG are presented in Figure 5. deviation to the arithmetic mean.
As the implemented coarse-graining method is based on the inviolable conservation laws -mass continuity between them -, it can be seen that the mean effective density in the studied domain is conserved through the scales and only the homogeneity of the field is altered. The heterogeneities of the effective density of the material are smeared-out much faster when considering a periodic microstructure at l CG /d = 1, while the quasi-periodic microstructures require higher coarse-graining scales for the density heterogeneities to smear-out at l CG /d = 4 (Figure 6 (b)).

Elastic Properties
Once the density fields are computed, manipulating the balance of mass at the fine and the coarser scale leads to the computation of the effective displacement fields that can be differentiated to determine the strain fields.
The balance of linear momentum allows the evaluation of the effective stress 23 (a) l CG /d = 1 fields. First, we aim to determine the behaviour of the material prior to damaging. For this purpose, we put the microstructure at the core of a rectangular specimen on which two tensile and one shear test simulations are conducted.
From the microscopic mechanical fields, coarse-grained displacements, strains and stresses can be evaluated for different l CG . Coarse-grained strain and stress couples in Ω 0 (obtained from the three tests) provide an evaluation of the nine components C ij of the effective stiffness tensor C, representative of the elastic behaviour, at each material point for the considered scales. It was observed that in the specimen coordinate system, the shear-extension coupling terms vanish; the reduced expression for the effective elasticity tensor written in (26) is thus adopted: Fields of the C 11 component of the effective elasticity tensor of the quasiperiodic type 1 microstructure at three different coarse-graining scales l CG are 24 presented in Figure 7; the heterogeneity of C 11 is shown to persist for large regularisation scales and thus the influence of the distribution of holes on the effective stiffness fields. From the computed effective elasticity tensors, both the anisotropy and homogeneity can be evaluated. As mentioned previously, the symmetry order of the studied periodic and quasi-periodic microstructures (6 and 5-fold symmetry respectively) are expected to lead to an equivalent isotropic response. From here, we aim to determine the scale from which the symmetry orders actually governs the elastic isotropy. To do so, the two-dimensional elastic anisotropy index a r -defined in [33] -is analysed at each material point for each observation scale. An explicit expression of a r as a function of C ij and S Cij can be written as follows: Where S Cij represent the components of the compliance tensor S C defined as S C = C −1 . a r takes the value of 0 in the case of perfect isotropy. Otherwise, a r increases as the anisotropy strengthens. The main advantage of a r over other anisotropy indices (Kube [34], Zener [35],...) is its direct applicability in 2D, for any symmetry type, and its direct evaluation from the elasticity tensor. In this study, a r is computed from the evaluated elasticity tensors C  the microstructure, interactions between the crack and the structure of the material are observed, especially in the quasi-periodic microstructures. The crack can be expected to follow the path that would allow the maximum dissipation of energy. For type 1 quasi-periodic distribution, "resilient patterns" (red circles in Figure 11(b)) impose the deviation of the crack. For the type-2 quasi-periodic distribution, kinking of the crack is present, and due to the high amount of elastic energy stored in the specimen before kinking (red circles in evolution of the crack paths and their corresponding tortuosity at different scales in the quasi-periodic microstructures are shown in Figure 11. Counter intuitively, one can observe prominent tortuosity of the crack path even at large scales. In fact, for the type 1 quasi-periodic microstructure, the crack tortuosity only drops of less than 6% at l CG /d = 10 while the type 2 crack path tortuosity drops around 9% from its microscopic value (Figure 11(a)).   Figure   12(a)). The contribution of the wavelengths around λ/d = 5 corresponding to the distance between zones of "resilient patterns" (Figure 11(b)) is conserved through the scales and is responsible for 80% of the crack deflection. For the type 2 quasi-periodic microstructure, one clearly observe the absence of uniquely conserved high amplitude wavelengths across the scales ( Figure   12(b)), except for λ/d = 2 − 3 that actually corresponds to the kinking spots.
The crack path, in comparison with the type 1 -except for the kinking spotsdoes not conserve the same wavelengths suggesting a more easily smoothed crack path. Next, we focus on the influence of the microscopic heterogeneity on the resistance and toughness fields at different coarse-graining scales, and we confront the wavelengths controlling the heterogeneities with the wavelengths present in the crack paths.
(a) l CG /d = 1  Figure 14 shows the stress-strain response of a material in the neighbourhood of a crack at a specific abscissa in the domain, at l CG /d = 1 which corresponds to l CG = d = 3mm. As the coarse-graining mesh size H is equal to 1mm, we find 6 different stress-strain responses of the elements corresponding to the discretization of the damageable zone. The same trend is found at different coarse-graining scales l CG (Figure 14(a)) where a unique relation between the local stress and local strain states does not exist. Instead, as we move further from the crack path, and due to the regularizing nature of the coarse-graining, softening persists yet starts at lower critical stress levels leading thus to non-unique relations between the stresses and strains. Non-locality of the fields is thus probable.
Moreover, when following the stress-strain response of the elements along the crack path, especially at low scales, and as the crack propagates through different patterns in the microstructure, the critical stress state reached before softening is found to differ from one position to another along the path. We define the maximum Rankine stress reached at each position of the crack tip as the fracture strength or simply strength denoted σ f , without abuse of language. The evolution of the critical Rankine stress at different scales of observation is plotted in Figure 15 for the periodic (a), quasi-periodic type 1 (b) and quasi-periodic type 2 (c). The below figures thus show the significant influence of the local differences inside the microstructure -distribution of holes -on the fracture strength of the effective continuum. As the crack gets trapped inside the holes, much higher loading is required for the crack re-initiation, this phenomenon is mainly observed on the periodic geometry at the smallest scales, while for the quasi-periodic geometries, not only crack trapping influence the critical stress state reached, but also the crack deflection and deviation around the special "resilient patterns".
The strength fields at different coarse-graining scales l CG are studied and both the average critical stress and its coefficient of variation over the crack path for each length scale l CG are plotted in Figure 16. It's clear that the critical stress σ f decreases when l CG increases. Moreover, a tendency stating l CG is found. Similar expressions relating the tensile strength σ f to the characteristic length l c , E and g c are found in [25,36,37,38] for gradient and non-local damage models. Kindly refer to Section 3.6 for more details about this finding.  In Figure 16, one observes higher strength for the periodic microstructure in comparison with the two quasi-periodic microstructures. A study on the evolution of the "homogeneity" of the strength is also conducted. Here, the coefficient of variation is evaluated by considering the squared differences of the values from the normalised values of a homogeneous response -allowing the study of the deviation of the strength surpassing the geometrical influences of the TDCB geometry and the loading conditions. Unlike the elasticity, the effective strength of the microstructures remains highly heterogeneous even at large l CG . From Figure 16 The periodic's COV σ f converges to 0.6%. Counter-intuitively, the coefficient of variation of the effective strength of the type 1 microstructure is lower than that of type 2 even though the path is more complex, but this may be caused by the presence of kinking in specific places leading to a huge increase of the loadings before the crack saps in comparison to the rest of domain where the crack path is smooth and straight.
Looking at Figure 15, it's hard to quantify both the microstructural effects and the influence of l CG on the fracture strength evolution in the material. For this purpose, the fracture evolution is studied in the frequency domain, and F F T analysis allows to display the wavelengths and amplitudes to better depict the interactions of the microstructures and the coarse-graining impacting the strength. Figure 17 compares the wavelength spectrum of the F F T analysis transformed from the fracture strength's deviation D σ f . The on the energy absorbed by the material points along the crack path. The effective toughness G d is defined as the energy to total failure evaluated as: allowing thus the measure of the energy dissipated per unit volume from the start of the loading (t = t 0 ) until the fracture of the specimen (t = t f ). S is the mesoscopic stress tensor andĖ is the mesoscopic strain rate tensor.
The evolution of G d at different scales of observation is plotted in figure 18 for the periodic (a), quasi-periodic type 1 (b) and quasi-periodic type 2 (c).
The below figures thus show the significant influence of the local differences inside the microstructure -holes distribution-on the overall dissipated energy along the crack path of the obtained continuum. As the crack gets trapped inside the holes, much higher energy is required for the crack re-initiation, this phenomenon is mainly observed on the periodic geometry at the smallest scales, while for the quasi-periodic geometries, not only crack trapping influence the energy dissipation, but also the crack deflection and deviation around the special "resilient patterns".
A similar analysis to the one in Section 3.5 is conducted: a study of the evolution of this toughness parameter G d at different coarse-graining scales followed by an F F T analysis to better understand the relationship between the microstructure and the effective toughness.
The average toughness G d is inversely proportional to the coarse-graining scale l CG (Figure 19(a)), and we find G d ∼ 1 l CG . Indeed, coarse-graining admits that the displacement, stress and strain fields on a material point actually depend on the state variables distribution in a neighbourhood of the point under consideration. The size of the neighbourhood is depicted by l CG . Here, we find G d ∼ 1 l CG and σ f ∼ 1 √ l CG . As previously stated, similar expressions relating the tensile strength σ f to the characteristic length l c , g c and E can be found in [25,36,37,38]. From [25], the critical value of the tensile strength in uniaxial traction is given by: Eg c 3l c This suggests the following: for a fixed toughness g c , the same relation between σ f and l CG holds between σ f and l c : σ f ∼ 1 √ lc . Ultimately, for an experimentally determined σ f , the relationship G d ∼ gc lc stands. We recall that the relations found in the literature hold only for uniaxial traction. We mention that no consensus on the relation linking E, σ f , l c and g c taking into account the different loading conditions, and/or specimen geometries can be found in the literature. To emphasize, we recall that all the results presented in this paper are found without any a priori on the material behaviour at the coarse-grained scales.
A study on the evolution of the "homogeneity" of the effective toughness is conducted. In Figure 19, we plot the evolution of the effective fracture toughness G d as a function of l CG for the three microstructures (a) and the evolution of the corresponding coefficient of variation COV G d defining the heterogeneity of the effective toughness field (b).
As compared to the strength σ f , one notices that the heterogeneity of the toughness field G d is higher than the one of the strength field at the same scales. Yet no stability of the coefficient of variation of this quantity is observed at the large scales. As long as the crack path is straight, both

Discussion and Concluding Remarks
This paper proposes a model-free coarse-graining method that does not require specific boundary conditions (as opposed to classical homogenisation schemes) and is indeed applicable to non-periodic structures presenting Quasi-brittle behaviour The coarse-graining method introduces a length scale which implies softening of the material -and thus an equivalency to a softening behaviour with a process zone, without any a priori on the behaviour at the larger scales. Plotting the stresses against the strains shows a typical response of quasi-brittle materials where a linear elastic region is followed by a non-linear region before softening. The notion of strength is thus notable.
Non-local effects Without any assumption on the material behaviour, the absence of a unique behaviour law that links the local variables, i.e., local strains and local stresses, is illustrated. The stress-strain history of the elements at different distances to the crack path is different suggesting thus non-locality of the behaviour.
Once the behaviour at larger scales was established from the microscopic data, failure analysis was led on different heterogeneous materials, and for this purpose, three microstructures were considered: the periodic hexagonal distribution of holes and two types of kite&dart Penrose distribution of holes: at the nodal positions (type 1) and on the centroids of the kites and darts (type 2). A study on the effective material properties was led followed by a failure analysis to further understand interactions between the cracks, the scales and the microstructures. Williams' series [39] should give more insight on the multiscale behaviour. We are currently exploiting the framework for a meso-macro analysis. Moreover, we are working on a multi-scale dynamic analysis of crack propagation inside the considered microstructures.