Engineering Failure Analysis 70 (2016) 157–176

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Engineering Failure Analysis

j ourna l homepage: www.e lsev ie r .com/ locate /engfa i lana l

Review

Damage process in heterogeneous materials analyzed by alattice model simulation

Gabriel Birck a,⁎, Ignacio Iturrioz a, Giuseppe Lacidogna b, Alberto Carpinteri b

a Department of Mechanical Engineering, Federal University of Rio Grande do Sul, Porto Alegre, RS, Brazilb Department of Structural, Geotechnical and Building Engineering, Politecnico di Torino, Turin, Italy

a r t i c l e i n f o

⁎ Corresponding author.E-mail address: gabriel.birck@ufrgs.br (G. Birck).

http://dx.doi.org/10.1016/j.engfailanal.2016.08.0041350-6307/© 2016 Elsevier Ltd. All rights reserved.

a b s t r a c t

Article history:Received 26 April 2016Received in revised form 21 July 2016Accepted 22 August 2016Available online 29 August 2016

Several materials of technological interest could be considered as heterogeneous and their ran-dom nature can be accounted to be the cause of the nonlinear behavior. The quantitative eval-uation of damage in materials subjected to stress or strain states has great importance due tothe critical character of these phenomena, which, at a certain point, may suddenly give rise tocatastrophic failure. In previous studies, Carpinteri and his coworkers have presented differentaspects of the damage process characterization in heterogeneous materials. Three of these as-pects demand our attention: (i) the brittleness number to measure the brittleness level of thestructure under investigation; (ii) the fractal dimension in which the damage process develops;and (iii) the global indexes obtained for the Acoustic Emission (AE) analysis. In the presentwork, a version of the discrete element method formed by bars is used to explore these con-cepts. A set of quasi-brittle material specimens is simulated and, when it is possible, the nu-merical results are compared with experimental data. Moreover, a discussion of the obtainedresults aids to better understand the behavior of this kind of materials, describing the numer-ical method as a viable tool to extract information from experimental tests on the damageprocess.

© 2016 Elsevier Ltd. All rights reserved.

Keywords:Heterogeneous materialsLattice discrete element modelAcoustic Emission techniqueSize effectDamage process

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1581.1. The concept of brittleness number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1581.2. Acoustic Emission techniques: relation between the b-value and the damage process evolution in heterogeneous materials . 158

2. Discrete element method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1592.1. Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1592.2. Non-linear constitutive model to account for anisotropic damage . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

3. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1633.1. The concept of brittleness number in the context of the lattice method . . . . . . . . . . . . . . . . . . . . . . . . . 1633.2. Acoustic Emission techniques: relation between the b-value and the damage process evolution in an heterogeneous material 167

3.2.1. Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1673.2.2. General results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1683.2.3. AE results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

4. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

158 G. Birck et al. / Engineering Failure Analysis 70 (2016) 157–176

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

1. Introduction

The process of damage in structures is a main topic in the solid mechanic area. Originally it was analyzed by Kachanov [1] andthen by Lemaitre [2], they are considered among the most important researchers that proposed models of damage suitable to beapplied with success, mainly in ductile material. This approach was used not only in metals but also in other kind of materialscharacterized by non-linear damage such as rubbers and others plastic with hyperelastic behavior [3–5].

Heterogeneous materials constitute a very large part of our natural environment as well as a substantial fraction of man-madeobjects: glasses, polymers and amorphous materials are among the vast array of examples [6]. On a larger scale, porous media,composites and suspensions can be also mentioned. The characterization of damage in this kind of materials is often governedby more than a single crack. The studies of a set of small fractures interacting at different scale levels have great importance tounderstand and simulate the behavior of these materials. Accounting each individual crack in the heterogeneous materials,assessing its influence on the structural response and ultimately on the structural failure, is not a task that can be approachedusing conventional methods of analysis in solid mechanics [7]. In this kind of materials, in which scale effects, anisotropic damage,and associative behavior among defects are likely to occur, the methods where the possibility to naturally represent the discon-tinuity could be an alternative. Among these methods, it is possible to include a version of the lattice method used in the presentwork. In several works developed by Carpinteri, concepts and models to analyze this kind of materials have been developed, someof them are discussed below.

1.1. The concept of brittleness number

It is known that the structural brittleness is related to the susceptibility of fractures to propagate in unstable condition. Thestructural brittleness can be measured by the material toughness and its interaction with the plastic collapse mechanisms. Anoth-er aspect that plays a very important role is the characteristic dimension of the structure. Carpinteri [8] proposed a dimensionlessparameter, the brittleness number s, that allows to measure the structural fragility,

s ¼ Kc

σpR1=2 ; ð1Þ

where Kc represents the material toughness in terms of the critical stress-intensity factor, σp represents the yield stress, and R is acharacteristic dimension of the structure. This number was used in several publications in the last three decades. Among others, itcould mention its applications in the biomechanical field [9].

1.2. Acoustic Emission techniques: relation between the b-value and the damage process evolution in heterogeneous materials

A promising method for a non-destructive quantitative evaluation of damage is the Acoustic Emission (AE) technique [10].Physically, AE is a phenomenon caused by a structural alteration in a solid material, in which transient elastic-waves due to arapid release of strain energy are generated. AE's are also known as stress-wave emissions.

AE waves, whose frequencies typically range from kHz to MHz, propagate through the material towards the surface of thestructural element, where they can be detected by sensors which turn the released strain energy packages into electrical signals[10–18]. Traditionally in AE testing, a number of parameters are recorded from the signals, such as arrival time, velocity, ampli-tude, duration, and frequency. From these parameters, damage conditions and localization of AE sources in the specimenscould be determined [19].

Using the AE technique, an effective damage assessment criterion is provided by the statistical analysis of the amplitude dis-tribution of the AE signals generated by growing microcracks. The amplitudes of such signals are distributed according to the Gu-tenberg-Richter (GR) law [11,20] N(≥A) ∝ A−b, where N is the number of AE signals with amplitude ≥A. The exponent b of the GRlaw, the so-called b-value, changes with the different stages of damage growth. While the initially dominant microcracking gen-erates a large number of low-amplitude AE signals, the subsequent macrocracking generates more signals of higher amplitude. Asa result, the b-value progressively decreases when the damage in the specimen advances: this is the core of the so-called “b-valueanalysis” used for damage assessment.

On the other hand, the damage process is also characterized by a progressive localization identified through the fractal dimen-sion D of the damaged domain. It may be proved that 2b = D [21–23]. Therefore, by determining the b-value, it becomes possibleto identify the energy release modalities in a structural element during the AE monitoring. In the theoretical extreme case D =3.0, which corresponds to b = 1.5, a critical condition in which the energy release takes place through small defects distributedthroughout the volume. For D = 2.0, which corresponds to b = 1.0, the energy release takes place on a fracture surface. In theformer case, diffused damage is observed, whereas in the latter case two-dimensional macrocracks are formed leading to the

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separation of the structural element. The b lower bound (bmin = 1) could be overcome in certain circumstances. Carpinteri et al.[16,20] discussed this topic in details, where experimental results are interpreted with an analytic model, showing that when amain macro-crack is generated it is possible justified values of b b1.0. It is also important to point out that Pradhan et al. [24],among others authors, show by using the Bundle Model that the b-value obtained in the simulated damage process will be be-tween 1.5, when the damage is incipient, and 0.5 near the collapse.

Regarding the numerical simulation, it is important to note that an alternative set of computational methods particularly suit-able for the simulation of the AE, introduced during the 1960s, did not use a set of differential or integral equations to describe themodel to be studied in the space domain. Different elements were introduced to analyze the particular considered materials, e.g.,particles or bars. The process is called by Munjiza [25] “Computational Mechanics of Discontinua” and it is now as an integral partof cutting-edge research in different solid modelling fields. As examples of this new type of approach we can cite:

• Models obtained with a discrete particles method, originally proposed by Cundall and Hart [26] and applied in Munjiza et al.[27], Brara et al. [28], Rabczuk and Belytschko [29,30] and Silling et al. [31].

• Models made of bars linked at their nodes, which are known as lattice models. Among others, it should be mentioned the workby Chiaia et al. [32], Chakrabarti and Benguigui [33], Krajcinovic [7], Rinaldi and Lai [34], and Rinaldi et al. [35], whose approachconstitutes a very interesting way to simulate the continuum, and provides qualitative information that sheds light on the frac-ture behavior of quasi-brittle materials such as concrete and rocks.

The lattice model used in the present work is a version particularly suitable for AE simulation, originally proposed by Riera[36]. He used the equivalent properties developed for a regular truss-like lattice model by Nayfeh and Hefzy [37]. Riera andother researchers extended the applications of the Lattice Discrete Element Method (LDEM) to model shells subjected to impul-sive loading [38,39]; fracture of elastic foundations on soft sand beds [40]; dynamic fracture [41]; earthquake generation andspread [42]; scale effect in concrete [43] and rock dowels [44,45]; the determination of static and dynamic fracture mechanics pa-rameters and crack growth simulation [46–49]. This method was also employed to simulate AE test and compare the numericalresults with experimental data [50,51].

In the present paper, different simulations using the Lattice Discrete Element Method (LDEM) will be adopted to explore theheterogeneous material damage process by using the concepts and methods briefly explained in the Introduction (1.1 and 1.2).

2. Discrete element method

2.1. Formulation

The Lattice Discrete Element Method (LDEM) used in this work represents the continuum by means of a periodic spatial ar-rangement of bars with the masses lumped at their ends. The discretization strategy, shown in Fig. 1, follows Nayfeh andHefzy [37]. This kind of method is usually called as “Central Force Model” in the physics community [52].

This method uses a basic cubic module constructed with twenty bars and nine nodes. Every node has three degrees of free-dom, which are the three components of the displacement vector in the global reference system.

The lumped mass at the nodes is specified in such a way that they add up the total mass of the module, mmodule = ρLc3. There-by, one half the module mass is assigned to the node at the center of the module, mcenter = 0.5ρLc3, while the other half of themass is distributed among the eight corner nodes. It is worth noting that, since neighbor modules share their corner nodes,these nodes account for the contributions of multiple modules after the complete model assembly.

In the case of an isotropic elastic material, the equivalent axial stiffness per unit length of the longitudinal elements (locatedalong module edges and connecting the nodes in the center of the modules) is shown in Eq. (2).

EAl ¼ AlE ¼ ΦEL2c ; ð2Þ

LcX

Y

Z

)b()a(

Fig. 1. Discretization strategy of the LDEM: (a) basic cubic module, and (b) prism formed using several cubic modules.

160 G. Birck et al. / Engineering Failure Analysis 70 (2016) 157–176

where Al is the cross section area of the element and E is the Young's modulus of the discretized solid. The function Φ = (9 +8δ)/(18+24δ),where δ=9ν/(4−8ν), accounts for the effect of the Poisson's ratioν [37,42]. Similarly, the stiffness for the diagonal elementswith area Ad is,

EAd ¼ AdE ¼ 2ffiffiffi3

p δΦEL2c : ð3Þ

The coefficient 2=ffiffiffi3

pin Eq. (3) accounts for the difference in length between the longitudinal and the diagonal elements, i.e.,

Ld ¼ 2=ffiffiffi3

pLc.

It is important to point out that for ν = 0.25, the correspondence between the equivalent discrete solid and the isotropic con-tinuum is complete. On the other hand, discrepancies appear in the shear terms for values of ν ≠ 0.25. These discrepancies aresmall and can be neglected in the range 0.20 ≤ ν ≤ 0.30. For values outside this interval, a different array of elements for thebasic module should be used (see [37]). Notice that this lattice model can not exactly represent a local isotropic continuummedia. It can also be argued that no perfect locally isotropic continuum exists in practical engineering applications. Isotropy insolids is a bulk property that reflects the random distribution of the constituent. A comprehensive study on the effect of theLDEM lattice geometry on the value of the Poisson's ratio can be found in Rinaldi et al. [35].

Newton's second law is enforced at every node to obtain the system of equations

M€x tð Þ þ C _x tð Þ þ F tð Þ−P tð Þ ¼ 0; ð4Þ

where vectors €xðtÞ and _xðtÞ contain respectively the nodal accelerations and velocities; M and C are the mass and damping ma-trices, respectively, the vectors F(t) and P(t) contain respectively the internal and external nodal forces. Since M and C are diag-onal, the equations are not coupled, and they can be easily integrated in the time domain using an explicit finite differencescheme.

It is worth noting that since nodal coordinates are updated at every time step, large displacements are accounted withoutextra computation because the model maintains the objectivity with finite displacement. At the same time, the LDEM has a nat-ural ability to model cracks. They can be introduced into the models as pre-existent features or as the irreversible effect of cracknucleation and propagation. Crack nucleation and propagation require non-linear constitutive models for material damage inorder to allow the elements to break when they reach a critical condition. The details about the formulation and implementationof these non-linear constitutive models are given in the next section.

2.2. Non-linear constitutive model to account for anisotropic damage

Rocha et al. [53], and more recently Kosteski et al. [48], introduced non-linear constitutive models to account the reduction inthe element load carrying capacity due to the irreversible effects of crack nucleation and propagation. The bilinear model forquasi-brittle materials proposed by Rocha et al. [53] is used in this work and it will be briefly presented next. The readers arereferred to the above references for further details.

The area under the force versus strain curve (the area of the triangle OAB in Fig. 2) is the energy density required to fracturethe area of influence of the element. Thus, for a given point P on the force vs. strain curve, the area of the triangle OPC representsthe reversible elastic energy density stored in the element, while the area of the triangle OAP is the dissipated fracture energydensity. Once the dissipated energy density equals the fracture energy, the element fails and loses its load carrying capacity.On the other hand, in the case of compressive loads the material behaves as linear elastic. Thus, the failure under compression

EA

P

O

A

C

Bε ε εp r

Damage energy,Udmg

Elastic strain energy, Uel

F

Fig. 2. Bilinear constitutive model with material damage. In terms of bar axial force vs. bar strain.

161G. Birck et al. / Engineering Failure Analysis 70 (2016) 157–176

load is induced by indirect traction. This assumption is reasonable for quasi-brittle materials for which the ultimate strength incompression is usually from five to ten times larger than that in tensile (see [54]).

The parameters for the constitutive model in Fig. 2 are (see [53]):

• Force, F: this is the element axial force as a function of the longitudinal strain ε.• Element stiffness, EiA: depending whether longitudinal or diagonal element is considered, EdA or ElA is adopted, see Eqs. (2) and (3).• Length, Lc: length of the LDEM basic cubic module (see Fig. 1).• Specific fracture energy, Gf: the fracture energy per unit area, which is coincident with the material fracture energy, Gc.• Equivalent fracture area, Aif: where i could be l or d depending whether a longitudinal or a diagonal element, respectively. Thisparameter enforces the condition that the energy dissipated by the fracture of the continuum is equivalent to that of its dis-crete representation. The energy dissipated by the fracture of a material sample of size Lc × Lc × Lc due to a crack parallel toone of its faces is,

Γ ¼ GfΛ ¼ Gf L2c : ð5Þ

The energy dissipated by the fracture of a LDEM module is,

ΓLDEM ¼ Gf cA 4ð Þ 0:25ð Þ þ 1þ 42ffiffiffi3

p δ� �� �

L2c ; ð6Þ

where the first term accounts for the four edge normal elements, the second term for the internal longitudinal element, and thethird term for the four diagonal elements. The factor 2ffiffi

3p δ in the third term is the stiffness ratio between the diagonal and the lon-

gitudinal bar, which is the quotient between Expressions (2) and (3). The coefficient cA is the scaling parameter to establish theequivalence between Γ and ΓLDEM, given by Eqs. (5) and (6), respectively. In the case ν = 0.25 is adopted, it results:

Gf L2c≅Gf

223

cA

� �L2c ; ð7Þ

from which the value cA ≅ 3/22 is obtained.

Finally, the equivalent fracture areas of longitudinal and diagonal elements are:

Afd ¼ 4

22L2c ; ð8Þ

and,

Afl ¼ 3

22L2c ; ð9Þ

for the diagonal and longitudinal elements, respectively.

• Critical failure strain, εp: The so-called critical strain εp, as illustrated in Fig. 2, is the maximum strain before the damage initi-ation. εp is a micro-parameter, i.e., a parameter that governs the constitutive law at the bar level. Young's modulus E, the stressintensity factor Kc and the critical stress εp are related by the classical fracture mechanic expression [55] given below:

Kc ¼ σpYffiffiffiffiffiffiπd

p¼ EεpY

ffiffiffiffiffiffiπd

p; ð10Þ

in which Y is a parameter that accounts for the influence of the boundary conditions and the orientation of the critical fissure ofsize d. If the behavior is assumedup to the initiation of rupture is linear, thenσp= Eεp and, recalling the equivalence betweenKc

and the specific facture energy Gf , we obtain the expression:

ffiffiffiffiffiffiffiffiffiGf E

q¼ EεpY

ffiffiffiffiffiffiπd

p: ð11Þ

In order to simplify Eq. (11), an equivalent length deq is defined as follows:

deq ¼ dπY2: ð12Þ

162 G. Birck et al. / Engineering Failure Analysis 70 (2016) 157–176

Substituting Eqs. (11) in (12), then:

deq ¼Gf

ε2pE: ð13Þ

Eq. (13) indicates that deqmay be regarded as amaterial property, since it does not depend on the discretization level, representing infact a characteristic length of the material (similar as the width of the plasticity region in the head of the fissure in the Dugdale model).About this topic, see a classical Fracture Mechanics book as Kanninen and Popelar [55]. The link between the critical size concepts intro-duced by Taylor [56] is evident. Isolating the critical strain in the Expression (13), one can obtained:

εp ¼ffiffiffiffiffiffiffiffiffiGf

deqE

s: ð14Þ

The brittleness number, proposed by Carpinteri [8] as a measure of the degree of ductility of the structure, can be rewrittenusing Eqs. (10), (11) and (14) in the context of LDEM formulation, as presented in Eq. (15). Recall that s not only accounts forthe material properties (Kc and σp) but also the size of the structure, represented by its characteristic length R.

s ¼ffiffiffiffiffiffiffideqR

s: ð15Þ

By the Eq. (15), the brittleness number can be defined as the square root of the ratio between the characteristic length of thematerial (deq) and the characteristic length of the structure (R). The structure behavior will be brittle if s → 0, consequently,deq ≪ R, or ductile if s → ∞, i.e., deq ≫ R. It should be emphasized that the crack geometry and the boundary conditions are em-bedded in the definition of deq.

To understand the meaning of the characteristic length, deq, it can be considered that during the damage process in a structurecomposed by a material with specific value of deq, if a crack with dimension aeq greater or equal to the value of deq is generated,this implies that the crack will propagate in an unstable way when the tensile stress in the crack is σp = Eεp. However, for this tohappen, the characteristic dimension of the structure R should be larger than the crack length, R N aeq, that is, R N deq.

• Limit strain, εr: it is the strain value for which the elements lose their load carrying capacity (point B in Fig. 2). The limit strain isexpressed in terms of the critical strain:

εr ¼ Krεp; ð16Þ

where,

Kr ¼Gf

Eε2p

!Afi

Ai

!2Li

� �¼ deq� � Af

i

Ai

!2Li

� �: ð17Þ

It is important to note that in order to guarantee the stability of the method, the condition Kr ≥ 1 must be satisfied. This con-dition is enforced by using the restriction Li ≤ Lcr on the element length with:

Lcr ¼ 2Gf

Eε2p

!Afi

Ai

!¼ 2 deq

� � Afi

Ai

!; ð18Þ

for both, the longitudinal, i = l, and the diagonal elements, i = d. The values for the element cross sectional areas, Ai, are inExpressions (2) and (3), while the equivalent fracture areas, Aif, are those in Eqs. (8) and (9).

Also notice that, in contrast to the usual practice in finite elements, the constitutive relationship in the LDEM is not only afunction of the material properties. The element constitutive relationship presented in Fig. 2 is defined in terms of parameters de-pending on material properties (εp, E, deq and Gf), model discretization (Lc, Aif) or both of them (EdA and εr). For this reason, whenthe discretization level is changed, the uniaxial constitutive law used in each bar must be modified.

A second interesting feature of the method is that although it uses a scalar damage law to describe the uniaxial behavior of theelements, the global model accounts for anisotropic damage, since the elements are orientated in different spatial directions.

The randomness of the model is introduced considering Gf as a random field with a Weibull density function characterized bya mean μ(Gf) and a coefficient of variation CV(Gf). It is also necessary to consider the spatial correlation function of this randomparameter. In the present version of LDEM, the correlation function is considered constant, meaning that in the interval of thecorrelation length Lcorr = 2Lc.

The introduction of small perturbations of the cubic arrangement, generated by small initial displacements of nodal points,should also result in small changes in the stiffness of the elements. Hence, it is herein assumed that the stiffness coefficients of

163G. Birck et al. / Engineering Failure Analysis 70 (2016) 157–176

the LDEM model remain unaltered by small perturbations of the mesh. Moreover, the linear response of the model should remainunaltered within the range of interest. Basically, it is assumed that the nodes in the perturbed model are displaced from their po-sition in a perfect cubic arrangement, defined by nodal coordinates (xn , yn , zn), as indicated in Eq. (19).

Table 1The inp

s

0.4

2.0

7.0

xn þ rx Lcð Þ; yn þ ry Lcð Þ; zn þ rz Lcð Þ� �

; ð19Þ

where rx, ry and rz are random numbers with a normal distribution, zero mean and coefficient of variation CVp. Lc denotes thelength of the longitudinal elements in the cubic cell. The CVp value that best fits the experimental evidence was determined bynumerical experimentation [57,58], ie, CVp = 2.5%. The introduction of this type of perturbation in the mesh is fundamental toimprove the model performance in modelled specimens submitted to compression stress. In Iturrioz et al. [50,57] more detailsabout the mesh perturbation are presented.

In the context of LDEM, if it has two models with different sizes, i.e., different R and different material properties, Gf, E and εp(these values define a deq) but both models have the same brittleness number s, both global responses will present the same me-chanical behavior. This subject will be discussed in the first numerical application presented in Section 3.

3. Applications

Simulations using the numerical method shown in Section 2 are presented below. The applications are related to the aspectdiscussed in the section 1.1 and 1.2 of the Introduction.

3.1. The concept of brittleness number in the context of the lattice method

With the aim to explore the link between the brittleness number, s, proposed by Carpinteri [8], and the discrete elementmodel, three sets of models with different brittleness numbers are simulated with LDEM. These simulations have the aim to verifythat keeping the same brittleness number, for all settings, the shape of the global response, i.e., stress-strain curve is similar.

Cubes are simulated by applying an uniaxial displacement prescribed at a low speed in the z direction. That is, the prescribeddisplacement applied does not induce dynamic effects during the simulation. Each set (s = 0.4, 2, 7) has 5 different input param-eters: (i) the toughness Gf, (ii) the elastic modulus E, (iii) the model discretization, (iv) the characteristic length deq, and (v) thespecimen length R. The first two are kept constant while the last three are modified, and four simulations for each study case arecarried out.

The input data adopted considered for all cases (Gf = 50 N/m, E = 35 GPa, CVGf = 50% and CVp = 2.5%) are presented inTable 1.

In Fig. 3, the final configurations are presented, and the applied boundary conditions are shown. The mesh was not perturbednear the border in order to minimize local distortion when the prescribed displacements are applied.

A brittle behavior is expected for s = 0.4. This behavior can be seen in Fig. 3, in which there is a predominance of broken bars(red bars) in the region where macrocracks are formed. Few damaged bars(orange bars) appears in the final configurations. Whenthe level of s grows, the number of partially damaged bars also grows, indicating that the global behavior of the model will beductile.

In Fig. 4, the global results in terms of normalized stress and strain values are presented. Notice that, the axes in Fig. 4 werenormalized, for a better response comparison, by the parameter α = (deq1/deqi)0.5, in which deq1 is the deq value for the Test 1 anddeqi is the deq value for the Test i. By observing this figure it is possible to notice:

(i) The set of 4 simulations for the same s presents practically the same global normalized stress vs. strain curves, i.e. if theinput parameters are changed but the s number is kept constant, the shape of the stress vs. strain curve is similar.

ut data and the internal parameters computed. fmaxn represent the bar peak force that is reached when the element strain is εp.

Test deq [m] R [m] Lc [m] Ri/R4 α Kr εp fmaxn [N]

1 0.90 6.00 0.25 10,000 1.00 2.45 3.982E−05 34,8422 3.60E−2 0.24 0.01 400.00 5.00 2.45 1.991E−04 278.73 9.01E−3 0.06 0.0025 100.00 10.00 2.45 3.982E−04 34.8424 9.01E−5 6.0E−04 2.5E−05 1.00 100.00 2.45 3.982E−03 0.03481 23.44 6.00 0.25 10,000 1.00 63.75 7.807E−06 68312 0.94 0.24 0.01 400.00 5.00 63.75 3.904E−05 54.63 0.23 0.06 0.0025 100.00 10.00 63.75 7.807E−05 6.84 2.34E−3 6.0E−04 2.5E−05 1.00 100.00 63.75 7.807E−04 6.8E−31 275.78 6.00 0.25 10,000 1.00 750.11 2.276E−06 24152 11.07 0.24 0.01 400.00 5.00 750.11 1.136E−05 15.93 2.76 0.06 0.0025 100.00 10.00 750.11 2.276E−05 2.04 2.76E−2 6.0E−04 2.5E−05 1.00 100.00 750.11 2.276E−04 2.0E−3

δz = δ

δz = 0.0

s = 0.4

s = 2.0

s = 7.0

Fig. 3. Final configurations of a section inside the sample obtained during the simulations. The boundary conditions are presented in the first picture. The brokenbars are indicated in red, in orange the bars with partial level of damage, and in gray the undamaged bars.

164 G. Birck et al. / Engineering Failure Analysis 70 (2016) 157–176

(ii) The curves with s = 0.4 present a very clear brittle behavior. On the other hand, when s = 7 the behavior of the model isclearly ductile. The transition from brittle to ductile typical behaviors occurs approximately when s = 2. Preliminarily, itcan be considered that s = 2 is the transition value between a brittle behavior (s b 2) and a ductile behavior (s N 2).

The results presented in Fig. 4 are useful to indicate that the global response is very sensible to the brittleness number, nev-ertheless more exhaustive studies should be carried out to show how the introduction of different disorder levels could perturbthis tendency.

Stre

ss [

Pa]

/ α

Strain [mm/mm] / α

Fig. 4. LDEM simulation results in uniaxial stress test (α = (deq1/deqi)0.5).

Strain [mm/mm] /α

Dis

sipa

ted

Ene

rgy

[Nm

] /α

s=0.4s=2.0s=7.0

0 1 20

500

1000

1500

2000

2500

3000

3500

x 10-4Strain [mm/mm] /α

Ela

stic

Ene

rgy

[Nm

] /α

0 1 20

500

1000

1500

2000

2500

3000

3500

x 10-4

s=0.4s=2.0s=7.0

Fig. 5. Elastic (left) and the dissipated (right) energy represented for the three analyzed cases. The values were normalized to facilitate the comparison.

165G. Birck et al. / Engineering Failure Analysis 70 (2016) 157–176

In Fig. 5, the energy balance vs. strain for the three typical simulations is presented. The black line represents the typical brittlebehavior (s = 0.4), the gray line the transition ductile-brittle behavior (s = 2), and the blue line the ductile behavior (s = 7). Thepresented results are coherent with the diagrams in Fig. 4, since in the brittle models the dissipated energy increases in an abruptway, showing an unstable situation when the body collapses. In the ductile model does not appear a very clear instant of rupture,the dissipated energy is distributed during all the damage process.

The experimental results from uniaxial tensile test in quasi-brittle material specimens allowed us to verify also that s = 2 isthe transition value from ductile to brittle behavior, as shown below. Van Vliet and Van Mier [59] performed tests on sandstonespecimens of different sizes, the samples were loaded with uniaxial tensile stress. The specimens layout, and the global resultsobtained during the experiment are shown in Fig. 6. The information obtained during the test, in terms of the elasticity modulus

Fig. 6. Specimens and force-deformation curves measured by Van Vliet and Van Mier [59] on sandstone samples shown above.

Table 2Results obtained in sandstone uniaxial tensile test by Van Vliet and Van Mier [59].

Spec. R [mm] Gf [N/m] σp s

A 50 76.7 0.82 5.29B 100 111.3 1.22 3.03C 200 93.8 1.01 2.38D 400 135.1 0.96 2.12E 800 143.3 1.04 1.42F 1600 93.2 0.96 0.88

E = 12.3 GPa.

166 G. Birck et al. / Engineering Failure Analysis 70 (2016) 157–176

E, toughness Gf, characteristic limit stress σp, and the characteristic length R, are synthesized in Table 2. By introducing thesevalues in Expression (1) the s parameter is computed and displayed in the same Table 2. The experiments carried out byCarpinteri and Ferro [60] by using concrete specimens of different sizes are also presented below. The specimen configuration,and the obtained global curves are shown in Fig. 7. The tests parameters, in terms of E, Gf, σp, R, and the correspondent s valueare presented in Table 3.

For the Van Vliet and Van Mier [59] experiments, represented in Fig. 6, it is possible to observe for the D specimen (s = 2.12)a sensible change in the behavior for s ≅ 2. For s values lower than 2, specimens E(s = 1.42) and F(s = 0.88), the global responseis brittle. Finally, for s values higher than 2 a clearly ductile global response, A(s = 5.29), B(s = 3.03) and C(s = 2.38), isobserved.

There are similar trends for the results obtained by Carpinteri and Ferro [60]. The specimen with characteristic length of10 mm is considered in the case where a transition behavior with s = 2.079 occurs. Moreover, for s values higher than 2,spec. 1(s = 3) and spec. 2(s = 2.94), a ductile behavior occurs, while for s values lower than 2, spec. 4 (s = 1.86) and spec. 5(s = 1.34), a typical brittle behavior is obtained.

With the aim to explore the meaning of the deq we accepted that the brittleness number s = 2 indicates the transition frombrittle to ductile behavior, and combining the Eqs. (12) and (15) when s = 2, it is possible to write:

d ¼ 4RπY2 ; ð20Þ

and assuming for the evaluation of Y the existence of a penny-shaped crack in a body with finite size [61], i.e.,

Y ¼ 2π

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisec

πd2R

� �s: ð21Þ

Replacing Eqs. (12) and (21) in the expression of s given by Eq. (15),

s ¼ffiffiffiffiffiffiπdR

r2π

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisec

πd2R

� �s: ð22Þ

Extension δ, µm

Forc

e F,

kN

Fig. 7. Force-displacement curves measured by Carpinteri and Ferro [60] on concrete samples (E = 35 GPa).

Table 3Results obtained in concrete uniaxial tensile test by Carpinteri and Ferro [60].

Spec. R [mm] Gf [N/m] σp s

1 25 147 4.79 3.002 50 257 4.56 2.943 100 236 4.37 2.0794 200 158 3.80 1.865 400 286 3.72 1.34

E = 35 GPa.

167G. Birck et al. / Engineering Failure Analysis 70 (2016) 157–176

and considering that s = 2,

ηdR

� �¼ d

Rsec

πd2R

� �� �¼ π: ð23Þ

In Fig. 8, the function ηðdRÞ is plotted together with the characteristic value, determined by the Eq. (23). In this figure, it is pos-sible to observe that for d/R values higher than 0.83 a ductile behavior is expected, and for d/R lower than 0.83 a brittle behavioroccurs. Recall that, R is the characteristic size of the structure and d is a material characteristic length (see the Expressions (10)and (11)), and assume that Y is given by the Expression (21). Therefore, if we have a fracture discontinuity in the structure withits size equals to d, this fracture will indicate a critical situation (see Eq. (10)). In the other case, it will not be possible to havebrittle behavior because there is not enough material to let the crack propagate in unstable way. Notice that, this relation alsodepends on which kind of solicitation is applied to the structure. In other words, there is a dependence between the characteristiclength R and the type of the boundary condition that supports the structure.

3.2. Acoustic Emission techniques: relation between the b-value and the damage process evolution in an heterogeneous material

3.2.1. Model descriptionIn the present application, three cube specimens are simulated considering standard concrete (quasi-brittle material) with dif-

ferent sizes (30, 60, and 90 mm). The specimens are submitted to uniaxial compression, imposing a prescribe displacement in thez direction, with low velocity (the prescribed displacement does not induce dynamic effects during the simulation). Two sets ofsimulations are carried out with and without lateral restriction in both ends (see Fig. 9). In Table 4, the input data adopted inthe simulation are presented. For all models, the same discretization level was used (the basic cubic module length (Lc) is 0.0025 m).

Fig. 9. Representation of boundary conditions for the model with (R) and without (L) lateral restriction.

0.5 0.6 0.7 0.8 0.90

1

2

3

4

5

d/R

η(d/

R)

η d d sec π d R R 2 R( ) [ ]( )=

Fig. 8. Function ηðdRÞ. Notice that the condition ηðdRÞ ¼ π occurs when d/R = 0.83, and in this case when s = 2.

Table 4Parameters adopted in the LDEM simulations.

μGfCVGf

CVp E deq ρ ν Lc Lcorr

50 N/m 50% 2.5% 35 GPa 0.0096 m 2400 kg/m3 0.25 0.0025 m 2×Lc

168 G. Birck et al. / Engineering Failure Analysis 70 (2016) 157–176

Therefore, the threemodelswith L=30mm, L=60mmand L=90mmwere constructedwith 12×12×12, 24×24×24 and 36× 36×36 modules, respectively.

The material toughness is defined as a random field characterized by themean value μGfand the variation coefficient CVGf

, present-ed in Table 4, considering theWeibull statistical distributiondefinedasW(Gf)=1−e(−(Gf/β)γ),whereβ andγdependon μGf

andCVGf. This

random field is also influenced by the correlation length (Lcorr) which is assumed to be equal to 2 × Lc. The cubic module has an imper-fection random field, using Gaussian distribution with mean zero and variation coefficient of CVp=2.5% of Lc.

The other parameters, that define the model, are deterministic: the density (ρ), the initial Young's modulus (E), the Poissonratio (ν) and deq. Notice that deq, E, Gf and the critical strain, εp are related to Expression (13).

3.2.2. General resultsThe main results are presented below. In Fig. 10 the results for global stress vs. global strain are presented for both boundary

conditions (with and without lateral restrictions). These results are valid until the peak of the global stress-strain curve.The stress peak decreases when the specimen length (L) increases, as presented in Fig. 10. These results characterize the size

effect, which is intrinsic for quasi-brittle materials as observed by Rios and Riera [43].The energy balance vs. the global strain, for the case 60L (L = 60 mm without lateral restriction), is presented in Fig. 11a. It is

possible to observe that a little quantity of energy (blue line) is dissipated by damage before reaching the peak load.In Fig. 11b, the dissipated energy is plotted for bars oriented in the load direction (red line) and normal to the load direction

(black line). Bars which are perpendicular to the load direction (NBar) dissipate energy during all the damage process. This dis-sipation is related to the first typical behavior in the damage process for the quasi-brittle specimens submitted to uniaxial com-pression, that is, several microcracks are distributed in all domain due to indirect tensile stress (see Fig. 12a). In Fig. 11b, there aretwo curves for NBar because there are bars in the x and y direction.

Near the peak load, the energy dissipated by the bars aligned to the load (PBar) begins to increase when the fracture mode IIappears and the friction becomes the main dissipation energy source (Fig. 12b illustrates this situation). In Fig. 11 is also present-ed the global stress vs. strain curve (presented out of scale in light gray) and the zone where the LDEM results have no physicalmeaning (solid shadow area).

Fig. 12a and b describe the two typical damage stages for quasi-brittle materials submitted to uniaxial compression as de-scribed by Gross and Seelig [62]. Initially, a global distribution of cracks in the same direction of the excitation appears due tothe indirect tensile stress induced orthogonally. At the second stage, diagonal macrocracks appear and the energy dissipated byfriction between the inclined fissures becomes important.

This version of LDEM cannot capture the energy dissipation due to friction and, therefore, the numerical simulation of speci-mens submitted to uniaxial compression only makes physical sense up to reach the peak load. However, it should be mentionedthat bending, shear, and uniaxial tensile damage processes can be represented by means of LDEM (pre and post peak-load) usingthe same input data.

0 0.5 1 1.5 2 2.5x 10

−3

0

10

20

30

40

50

60

70

Strain [mm/mm]

Stre

ss [

MPa

]

30R60R90R

0 0.5 1 1.5 2 2.5x 10

−3

0

10

20

30

40

50

60

70

Strain [mm/mm]

Stre

ss [

MPa

]

30L60L90L

)b()a(

Fig. 10. Global stress-strain curve for the three specimens analyzed: (a) with lateral restrictions in the ends and (b) without lateral restrictions in the ends.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2x 10-30

2

4

6

8

10

12

Strain [mm/mm]

Ene

rgy[

Nm

] StressKineticElasticDissipated

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x10-3

Strain [mm/mm]

Ene

rgy

[Nm

]

StressPBarNBar

)b()a(

Fig. 11. Curve that corresponds to the size L = 60 mm for the case without restriction in both ends (60L): (a) energy balance during the damage process and (b)dissipated energy for bars oriented in the same load direction (PBar) and normal to the load direction (NBar), notice that two curves of NBar appear because it hasthe bars in the x and y directions.

169G. Birck et al. / Engineering Failure Analysis 70 (2016) 157–176

Comparison between numerical and experimental results for concrete specimens subjected to uniaxial compression load waspresented in Iturrioz et al. [50,57], Rios and Riera [43], and Miguel et al. [44].

Final configuration results for specimen with L=90 mm are presented in Fig. 13a, b, and c for the case without lateral restric-tion in both ends (90L) and it can be compared to the final experimental configuration in similar specimen presented in Fig. 12c.In Fig. 13d and e, two views for the case with lateral restriction in the ends (90R) are presented. In Fig. 13f, two experimentalconfigurations of similar specimens for lateral restriction in both ends are presented for comparison. In Fig. 13a to e, red, orangeand gray colors represent broken, damaged, and undamaged bars, respectively.

Comparing experimental and numerical results, it is possible to see that the results are coherent, i.e., the same final fracturemechanism clearly appears.

3.2.3. AE resultsAcoustic Emission results for the simulated cases are presented in this section. In Fig. 13a and d, the positions (S) on the spec-

imen surface where the acceleration is registered (simulating an AE device) are indicated. The acceleration was measured in aperpendicular direction to the surface.

Comparison between experimental and numerical AE event amplitude (A) was carried out in both time (t) and frequency (f)domains, as can be seen in Fig. 14. From these figures, it can be noticed that the results in the time and frequency domain aresimilar.

In Fig. 15a, accumulated (Nacum) and instantaneous number (N) of AE events are presented versus the normalized time. InFig. 15b, it also presented the logarithm of the AE amplitude vs. the normalized time.

A comparison between cases without (60L) and with lateral boundary restriction (60R) are presented in both figures. It can benoticed for the 60L case, that there are no AE events between 0.3 and 0.7 of normalized time. In both cases, there is a great quan-tity of low AE events at the beginning and then the number and amplitude of the events increase at the end of the simulation. Theglobal stress curve was included (out of scale) in these figures.

)c()b()a(

Fig. 12. Schemes that illustrate the damage process in quasi-brittle materials under compression: (a) cracks appear in loading direction; (b) self-organization andpropagation at an angle and the sliding dissipation process begins to be dominant; and (c) experimental results for the case without lateral restriction in both ends.

XZ

z= '

z=0

S

S

XY

X

Y

Z

(a) (b) (c)

XZ

z= ' , x= y=0

x= y= z=0

S

XY

S

XZ

(d) (e) (f)

Fig. 13. Final configuration for 90L (a, b and c) and 90R (d and e) models; (a) and (d) illustrate the boundary conditions in each simulation. In (f), it is presentedan experimental result with lateral restriction in the ends.

170 G. Birck et al. / Engineering Failure Analysis 70 (2016) 157–176

The b-value, which is the angular coefficient in the relation between log(Nacum b m) vs. m(m= logA), can be evaluated, forthe specimen 60L, through Fig. 16a for the initial stage of the damage process. It can be noticed that this curve presents a linearbehavior. Moreover, in Fig. 16b, a typical result obtained in experimental test by Carpinteri et al. [19,20] is presented; here, the b-value is usually within the interval [1, 1.5].

Finally, Fig. 17a presents the variation of the b-value during all the process simulated for the specimen 60L (line with squaremarker) and the specimen 60R (line and circles). Fig. 17c exposes three experimental curves for uniaxial compression tests carriedout by Carpinteri et al. [63].

As indicated in Section 1, it is possible to relate the dimension of the fracture distribution D to the b coefficient (D = 2b). Con-sidering the experimental evidence [16,20], it was also pointed out that the b-value can be less than one. This is possible when thebody test present a main growing macro-crack. This can be seen in Fig. 17a, for the sample 60L, where the minimum b-value isb1.0, and in the Fig. 17c, that presents experimental results obtained by Carpinteri et al. [63].

1.6x10-3s 1.6x10-3s

Experimental Numerical

)b()a(

Fig. 14. Experimental and numerical comparison: (a) amplitude (A) vs. time (t); (b) amplitude (A) vs. frequency (f) obtained by a numerical simulation.[51]

0 0.2 0.4 0.6 0.8 1 1.2t/tmax

NNacumStress

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

(15*

N),

Nac

um

60L

NNacumStress

t/tmax

60Rx103

0

1

2

3

4

5

6

(15*

N),

Nac

um

x103

(a)

0 0.2 0.4 0.6 0.8 1 1.24

6

8

10

12

14

t/tmax

m

mStress

0 0.2 0.4 0.6 0.8 14

6

8

10

12

14

t/tmax

m

60R60L

mStress

(b)

Fig. 15. Results for the specimen L = 60 mm with and without lateral restriction: (a) accumulated and instantaneous numbers of AE events vs. normalized time.(b) m = log(AE amplitude) vs. normalized time. In all plots the global stress vs. normalized time out of scale are presented.

171G. Birck et al. / Engineering Failure Analysis 70 (2016) 157–176

Fig. 17b presents the spatial distribution of the AE events sources in different intervals during the damage process. It is inter-esting to compare different regions of the Fig. 17a with the position of the source of the AE during the simulation.

Excluding the less significant initial stages of the loading process [0.0, 0.2] t/tmax, sensitive variations of the signals' amplitudestake place mainly in the interval [0.80, 1.15] of normalized time for the 60L specimen. While, for the 60R specimen, damage takesplace with almost constant signals emission amplitudes. For this last specimen, only in the interval [0.82, 1.0] significant changesin the AE signals amplitudes occur (see Fig. 15b).

This means that for the specimen 60R, subjected to lateral restrictions in the ends, damage proceeds with the comminution ofthe entire bulk of the material in numberless similar sized particles. This phenomenon is characterized by AE's signals of nearlyconstant amplitude. While the damage of the 60L specimen, free of restrictions in the lateral ends, is characterized by the forma-tion of preferential large fracture surfaces, predominantly formed towards the end of the loading process [20,64].

Therefore, in the numerical model the b-value trends for the specimens 60L and 60R reflect this situation. Because the fractaldimension of the damage is given by the relation D = 2b, though the spasmodic evolution of the damage is reflected in the os-cillating behavior of the b-value evolution, at the end of the loading process it can be seen as for specimen 60L damage is prev-alent localized on a macro-fracture (with b around 1.0, D is about 2.0). For the specimen 60R, instead, the damage is distributed inthe entire bulk of the material (with b around 1.5, D is about 3.0), (see Fig. 17a).

To further proof of this behavior, it can be also observed as during the simulated damage process for the specimen 60L, thespatial distribution of the bars that produce AE events has been mainly localized in the interval [0.80, 0.95]. While, for the spec-imen 60R, the distribution of the bars producing AE events are uniformly distributed in the bulk of the material throughout all theloading process (see Fig. 17b).

4. Conclusions

In the present work three aspects of the damage process characterization are analyzed, using a version of the Lattice DiscreteElement Method (LDEM): the brittleness number to measure the brittleness level of a structure under investigation, the fractal

4.5 5 5.50

0.5

1

1.5

2

2.5

Log

N(<

A)

M

= 1.9b

100 101 102 103 104 105

A (µV)100 101 102 103 104 105

A (µV)100 101 102 103 104 105

A (µV)

106

105

104

103

102

101

100

104

103

102

101

100

104

103

102

101

100

N(A

E E

ven

ts)

N(A

E E

ven

ts)

N(A

E E

ven

ts)

I

1

b=1.75

1 1b=1.39

b=1.26

II III

)b()a(

Fig. 16. Results in terms of b-value: (a) measured in the normalized time interval [0.4, 0.7] for the test 60 L; (b) typical experimental result obtained in uniaxial compression test [19].

172G.Birck

etal./EngineeringFailure

Analysis

70(2016)

157–176

0.2 0.4 0.6 0.8 10.5

1

1.5

2

2.5

3

3.5

4

t/tmax

-val

ueb

60R60L

(a)L[0.08,0.45] L[0.80,0.95] R[0.08,0.3] R[0.3,0.80] R[0.82,0.85] R[0.85,0.92]

XZ

XY

XZ

XY

XZ

XY

XZ

XY

XZ

XY

XZ

XY

(b)

0 0.2 0.4 0.6 0.8 10 0.2 0.4 0.6 0.8 10 0.2 0.4 0.6 0.8 1t/tmax

b -va

lue

0

0.5

1

1.5

2

2.5

3

0

0.2

0.4

0.6

0.8

1

P/P

max

Peak load = 7.12 kN

b-value = 1

S1a(30 mm)0

0.5

1

1.5

2

2.5

3

b -va

lue

0

0.2

0.4

0.6

0.8

1 1

P/P

max

Peak load = 24.60 kN

b-value = 1

S2a(60 mm)0

0.5

1

1.5

2

2.5

3

0

0.2

0.4

0.6

0.8

b -va

lue

P/P

max

Peak load = 96.59 kN

b-value = 1

S3a(120 mm)

t/tmax t/tmax

(c)

Fig. 17. (a) Evolution of b-value during all the damage process; (b) the intervals and spatial distribution of the bars that produce the AE events during the damageprocess simulated; and (c) experimental results carried out by Carpinteri et al. [63] in similar specimens.

173G. Birck et al. / Engineering Failure Analysis 70 (2016) 157–176

dimension in which the damage process is develops, and the global indexes obtained for the Acoustic Emission analysis. Simula-tions were carried out using the LDEM and the following conclusions are pointed out:

• When the LDEM formulation was presented, it was established that the uniaxial constitutive law used to define the uniaxial be-havior must define three of a set of four material parameters: the longitudinal Elastic Modulus E, the toughness Gf, the criticalstrain εp, and a material characteristic length deq. Using the brittleness number (s) produced a better understanding of themeaning of the material characteristic length deq. This means that s could be defined as the square root of the ratio betweendeq and the structure characteristic length R, as presented in Expression (15).

174 G. Birck et al. / Engineering Failure Analysis 70 (2016) 157–176

• In the specimens simulation of quasi-brittle material submitted to uniaxial tension with different material parameters and size,maintaining the parameter s constant, it was possible to verify that for equal values of s the global response of the specimenswas similar. Since we had a typical ductile response when s N 2, and a typical brittle behavior when s b 2, a change in the globalresponse occurred when s = 2. For further boundary conditions, the critical value for s must be investigated.

• It was possible to observe the same behavior by computing the value of s in two series of experimental results made by VanVliet and Van Mier [59] and Carpinteri and Ferro [60].

• It was possible to recognize that if in a specimen submitted to uniaxial tensile load we have a crack with length a, characterizedby a value aeq = Y2πa equal or greater than deq, fracture becomes unstable only if deq is lesser than the structural characteristiclength R, and when the crack neighborhood is reaching the critical deformation εp. Otherwise, if deq is similar to R it is physicallyimpossible for the crack to propagate.

• It was possible to observe that, if we change the loading conditions, the way to compute the characteristic length must bechanged.

• The experimental analysis of the b-value evolution during the damage process, and its correlation with the damage spatial dis-tribution obtained by the numerical model presented a good coherence. It was confirmed that the b-value decreases when theAE are emitted from a concentrated region, while the b-value increases when the damage is generated from a more distributedregion. The possibilities to link the b-value with its spatial distribution during the numerical simulation process could be a po-tential tool to be used in more complex geometries.

Nomenclatureβ and γ Scale and shape parameters in the Weibull distribution functionεp Critical failure strainεr Limit strainAl and Ad Cross-sectional areas of the longitudinal and diagonal elements respectively in the lattice model usedAlf and Ad

f Equivalent fracture area of the longitudinal and diagonal elements, respectivelyd Critical crack sizedeq Characteristic length of the materialKr Failure coefficient that relates the critical strain εp with the limit strain εr (εr = Krεp)Lcr Value of the elemental length Lc when Kr = 1Y Dimensionless parameter that depends on both the specimen and crack geometryα Parameter that relatesΓ Dissipated energy by the fracture of a solid cubeΓLDEM Dissipated energy by the fracture of a cube into two parts in the lattice model usedF(t) and P(t) The internal and external nodal load vectors UdmgDamage energyM and C Mass and damping matricesx; _x and €x Vectors that contain the nodal displacements, velocities and accelerations, respectivelyμ Mean valueν Poisson's ratioΦ,ν Coefficients that let to compute the transversal area of the elements and the length Lcρ Densityσp Yield stressA Acoustic Emission (AE) amplitudeb Called “b-value”, is the negative slope of the logN(≥A) vs. m relationcA Scaling parameter to establish the equivalence between Γ and ΓLDEMCV Coefficient of variationD Fractal numberE Young's modulusElA and Ed

A Axial stiffness per unit length of the longitudinal and diagonal elements, respectivelyF Axial forcef Frequencyfmax Maximum forceGc Material fracture energyGf Specific fracture energyKc Material toughnessL Size of the modelLc Basic cube length of the lattice model usedLd Diagonal elements lengthLcorr Correlation lengthm Acoustic Emission amplitude A in logarithm scale (m = log10A)mmodule Lumped mass at the nodes of the basic cubemmodule Lumped mass of the center nodeN Instantaneous number of AE events

175G. Birck et al. / Engineering Failure Analysis 70 (2016) 157–176

N(≥A) Number of Acoustic Emission (AE) signals with amplitude ≥ANacum Accumulated number of AE eventsR Characteristic dimension of the structurerx,ry and rz Random numbers with a normal distributions Brittleness numbert Timetmax Maximum timeUdmg Damage energyUel Elastic strain energyW Weibull type probability distribution functionxn,yn and zn Nodal coordinates

Acknowledgements

The authors wish to thank the National Council for Scientific and Technological Development (CNPq - Brazil) and Coordinationfor the Improvement of Higher Education Personnel (CAPES - Brazil) for funding this research.

References

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