32nd EWGAE 35

Czech Society for Nondestructive Testing

32nd European Conference on Acoustic Emission Testing

Prague, Czech Republic, September 07-09, 2016

A VERSION OF THE DISCRETE ELEMENT METHOD

IN THE SIMULATION OF THE ACOUSTIC EMISSION TESTING

Gabriel BIRCK1, Ignacio ITURRIOZ1

1 Department of Mechanical Engineering, Federal University of Rio Grande do Sul, Porto

Alegre, RS, Brazil. Phone: +55 51 3308 3529; e-mail: gabriel.birck@ufrgs.br, ignacio@mecanica.ufrgs.br

Abstract

In the present work, events of Acoustic Emission testing in quasi-brittle material were simulated by a version of the

discrete element method. In this numerical approach the solid is modelled by means of a periodic spatial

arrangement of bars with the masses lumped at their ends. The results obtained by numerical approach are

evaluated by Acoustic Emission parameters, as b-value, RA value and average frequency, and compared with the

moment tensor analysis. The results showed coherent between both forms of evaluation. The numerical approach

was able to simulate Acoustic Emission events and provide more information about the damage process.

Keywords: Lattice Discrete Element Method, Damage evaluation, Acoustic Emission, Moment Tensor

1. Introduction

From a physical point of view, damage phenomena consist either of surface discontinuities in

the form of cracks or of volume discontinuities in the form of cavities [1, 2]. Macroscopically,

it’s, therefore, necessary to identify internal variables that reflect the damage level in the

material. The most advanced method for a non-destructive quantitative evaluation of damage

progression is the Acoustic Emission (AE) technique. Physically, AE is a phenomenon caused

by a structural alteration in a solid material in which transient elastic-waves, due to a rapid

release of strain energy, are generated. AEs are also known as stress-wave emissions.

In the present work a version of the Discrete Element Method built by bars (LDEM) are employed. This

approach belongs to the alternative set of computational methods called by Munjiza [3], "Computational

Mechanics of Discontinua" introduced during the 1960s, which is characterized by the lack of

differential or integral equations to describe the model to study the space domain. In the mentioned

approach, the behaviour of the solid is a function of the individual elements, e.g., particles or bars.

Here, a notched beam subjected to the three-point bending test is simulated with LDEM.

The beam is built in quasi-fragile material. Global stress-strain is evaluated and accelerations

in some points of the specimen’s surface are registered. The acceleration registered is

interpreted as data provided by a real device EA.

2. Acoustic Emission

One promisor method for a non-destructive quantitative evaluation of damage progression is

the Acoustic Emission (AE) technique. Physically, AE is a phenomenon caused by

a structural alteration in a solid material, in which transient elastic-waves, due to a rapid

release of strain energy, are generated. AEs 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 the structural element, where they can be detected by sensors which turn the

released strain energy packages into electrical signals [4, 5]. Traditionally, in AE testing, a number

of parameters are recorded from the signals, such as arrival time, velocity, hits, count, rise time,

amplitude, duration and frequency. From these parameters damage conditions and localization of

AE sources in the specimens are determined Carpinteri et. al. [6].

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2.1. Damage process evolution

An effective damage assessment criterion is provided by the statistical analysis of the

amplitude distribution of the AE signals generated by growing microcracks. The amplitudes

of such signals are distributed according to the Gutenberg-Richter (GR) law [4, 7], N( ≥ A) ∝

A-b, where N is the number of AE signals with amplitude ≥A. The exponent b of the GR law,

the so-called b-value, changes with the different stages of damage growth. While the initially

dominant microcracking generates a large number of low-amplitude AE signals, the

subsequent macrocracking generates more signals of higher amplitude. As a result, the b-

value progressively decreases when the damage in the specimen advances, as shown in Figure

1. This is the core of the so-called “b-value analysis” used for damage assessment.

Figure 1. The b-value evolution in experimental test [7]

2.2. Crack classification

The shape of the AE waveforms characterizes the fracture mode [8]. So, to classify active cracks

parameters as Rise Angle (RA) and average frequency are calculated from AE events. The RA

value is defined as the ratio between the rise time and the maximum amplitude. The average

frequency is obtained from the relation of AE ring-down count and the duration time of the signal,

as shown in Figure 2. The AE ring-down count corresponds to the number of threshold crossings

within the signal duration time [9]. From these two parameters, cracks are readily classified into

tensile and shear cracks as illustrated in Figure 2.

The tensile mode of crack (Mode I), which includes opposing movement of the crack sides,

results in AE waveforms with short rise time and high frequency. On the other hand, shear type of

cracks (Mode II) usually result in longer waveforms, with lower frequency and longer rise time.

(a) (b)

Figure 2. (a) Typical waveforms for tensile and shear events, (b) Relationship between average frequency and

RA value

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2.3. Moment tensor theory

By the generalized theory of AE [10], elastic displacement u(x,t) at location x due to the crack

displacement in y at the time t is represented as [11],

��(�, �) = ���,�(�,�, �)��� ∗ �(�), (1)

where ���,�(�,�, �) represents the spatial derivatives of Green’s functions, S(t) is the kinetic

source (time function of crack motion), Mpq is the moment tensor components (kinematics of

crack motion) and (*) represents the convolution function. The moment tensor is a symmetric

second-rank tensor and is comparable to the elastic stress in elasticity (see Figure 3a), which

is expressed as [11-13],

��� = ���������∆�, (2)

where Cpqkl is the constitutive tensor, l is the direction vector of crack motion, n is the normal

of the crack surface and ∆V is the crack volume.

If only the first motion of the P-wave A(x) is taken into account and assuming that all of the

moment tensor components have the same time-dependency, Equation (1) can be simplified

as [11-15],

�(�) = ����(�,�)� [�1 �2 �3] ��11 �12 �13�21 �22 �23�31 �32 �33� ��1�2�3�, (3)

where Cs is the physical coefficient containing the sensor sensitivity, obtained by pencil-lead

break test [16]; R is the distance between the crack y and the detection point x, and r1, r2, r3

are its different direction cosines, respectively; Re(t,r) is the reflection coefficient between

vector r and the direction t of the sensor (see Figure 3b). In the case where the P-wave arrives

vertically to the surface Re(t,r)=2. Equation (3) leads to a series of algebraic equations in

which the components of the moment tensor Mpq are unknowns, where there are 6

independent components.

(a) (b)

Figure 3. (a) Elements pf the moment tensor, (b) Crack nucleation and AE detection

The source location is determined from the arrival time differences (ti) between the

observation xi and xi+1, solving the equations,

�� − ��+1 = |�� − �|− |��+� − �| = ���� , (4)

where vp is the velocity of P-wave, assumed constant.

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By the analysis of eigenvalues and associated eigenvectors of the moment tensor it is possible

to classify the crack (failure mode), and obtain the crack motion vector (l) and the crack

normal vector (n) [11]. The eigenvalue is decomposed into a shear component X, a deviatoric

component Y and a hydrostatic component Z, which, X is the maximum shear contribution, Y

is set as the maximum deviatoric tensile component and Z is the maximum isotropic tensile

component. Three eigenvalues are normalized and decomposed as follows:

� + � + � = 1 = �1 �1⁄ ,

0− � 2⁄ + � = �2 �1⁄ , −� − � 2⁄ + � = �3 �1⁄ ,

(5)

where λ 1 , λ 2 and λ 3 are maximum, intermediate and minimum eigenvalues, respectively.

The cracks are classified, as proposed by Ohtsu [11], into three types of AE sources: X > 60%

as shear cracks; X < 40% as tensile cracks, and 40% < X < 60% as mixed-mode cracks.

3. Lattice Discrete Element Method (LDEM)

The version of the truss-like Discrete Element Method (DEM) proposed by Riera [17], which

is used in this paper, is based on the representation of a solid with a cubic arrangement of

elements which is only able to carry axial loads. The cubic module is presented in Figure 4a

and Figure 4b. The mass is lumped at the nodes. Each node has three degrees of freedom,

corresponding to the nodal displacements in the three orthogonal coordinate directions.

The equations that relate the properties of the elements to the elastic constants for an isotropic

medium are presented in Equation (6). � =9�

4 − 8� ,��� = ���2 (9 + 8�)

2(9 + 12�),��� =

2√3

3��, (6)

where � and � denote Young’s modulus and Poisson’s ratio, respectively, while �� and ��

are the areas of the normal and the diagonal elements and �� is the size of the basic cubic

module. The resulting motion equations, obtained with this spatial discretization, can be

written in the well-known form presented in Equation (7). ��̈(�) + ��̇(�) + ��(�)− �(�) = 0, (7)

where � is the vector of generalized nodal displacements, � is the diagonal mass matrix, � is the

damping matrix, also assumed diagonal, �� is the vector of the internal forces acting on the nodal

masses and � is the vector of external forces. The operator (•)̇ denotes the time (�) differentiation �(•)/��. If � and � are diagonal, the Equation (7) is not coupled and, therefore,

the explicit central finite difference scheme can be used to time domain integration. Since the

nodal coordinates are updated for each time step, large displacements can be accounted in

a natural and efficient manner. This article adopts a relationship between axial force and axial

strain in the uniaxial elements (bar), based on the bilinear law proposed by Hillerborg [18], which

is presented in Figure 4c. The specific fracture energy (Gf) is directly proportional to the area

below the bilinear constitutive law. Another important feature of this approach is the assumption

that Gf is a 3D random field with a Weibull probability distribution.

The local strain associated with maximum loading in each bar is called critical strain (��).

This value is also a random variable and its variability, which is measured using the

coefficient of variation CV, is related to the Gf parameter by the Equation (8), ���� = 0.5���� . (8)

32nd EWGAE 39

The minimum value of ��, determined for each specimen bar, is associated with the global

strain for which a specimen loses linearity.

Mesh perturbation was incorporated to improve the performance when the body is submitted

to compression. It is possible to observe in Figure 4c, that the bar failure occurs only when

submitted to tensile solicitation. In compression, the bars have linear elasticity behaviour [19].

More exhaustive explanations of this version of the lattice model can be found in Kosteski et

al. [20, 21]. Applications of the LDEM in studies involving non-homogeneous materials

subjected to fracture, such as concrete and rock, can be found in Riera and Iturrioz [22],

Dalguer et al. [23], Miguel et al. [24], Iturrioz et al. [25] and Miguel et al. [26].

(a) (b) (c)

Figure 4. LDEM: (a) basic cubic module, (b) generation of a prismatic body, and (c) Bilinear constitutive model

with material damage

3.1. AE tests simulated by LDEM

The numerical approach employed in this paper was used to simulate Acoustic Emission tests

in others cases, as presented in Iturrioz et al. [27], where comparison between experimental

and numerical tests performed. In terms of the distribution of AE event amplitudes in time

and frequency, the results were seen to be consistent, as can be seen in Figure 5.

Figure 5. Comparison between numerical and experimental results [27].

4. Applications

A plain concrete beam, with pre-notched and depth equal to half the overall height, was

simulated to a three-point bending (TPB) with LDEM [28]. Standards results as force and

vertical deflection were computed and AE test was performed. The scheme and main

geometrical parameters are reported in Figure 6. The material properties are: Young’s

modulus 35GPa; Poisson’s ratio 0.25; density 2400 kg/m3, and specific fracture energy (Gf)

70 N/m. The beam was discretized with 168x20x20 basic cubic with length (Lc) equal 5 mm.

Twelve sensors were used to EA test, two sensors per side.

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Figure 6. Schematic representation of TPB test

The statistical analysis of the amplitude distribution (b-value) for three periods of the time is

presented in Figure 7. In Figure 8, the time distribution of instantaneous and accumulated AE

events are plotted, also the reaction (force) by the time.

Figure 7. b-value distribution for three periods

Figure 8. Instantaneous and accumulated AE events, and force by the time

The b-value remains low and about constant for the periods analysed, as can be seen in Figure

7, because the AE are mainly emitted from a specific region of the structure (see Figure 9),

which is expected since the beam has a notch. By the Figure 8, it’s observed that AE events

occur throughout all the period. For the period I, shown in Figure 9, the damage is more

widespread than in others, therefore, the b-value is larger.

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Figure 9. Damage distribution in the three periods of b-value analysis

Crack classification for three AE events were performed by AE parameters and by moment

tensor analysis. In Figure 10, crack classification by AE parameters, RA value and Average

Frequency was performed by two sensors. These events were chosen because the amplitude of

the first motion of the P-wave and the beginning of the event are clearly defined.

Figure 10. Crack classification by AE parameters

In Figure 11, it is represented the position and classification for the AE events analyzed by

tensor moment analysis. The arrows represent the crack motion vector (l) and the crack

normal vector (n), which are interchangeable [29]. In Table 1, the moment tensor normalized

is presented. Through these figures, the classification by AE parameters and moment tensor

are coherent, that is, the same classification was obtained. It is noteworthy that using moment

tensor analysis it is possible get the orientation and displacement vector of the crack.

Figure 11. AE events representation

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Table 1. Normalized moment tensor for each event

Event Normalized moment tensor

A

X=10.40% � ���max������ = �0.02 0.04 0.10

0.04 0.07 0.37

0.10 0.37 1.00

�

B

X=32.36% � ���max������ = �0.00 0.01 0.09

0.01 1.00 0.32

0.09 0.32 0.33

�

C

X=57.28% � ���max (���)

� = �0.07 0.02 0.13

0.02 0.90 1.00

0.13 1.00 0.27

�

5. Conclusions

The b-value evolution in the damage process is related with the spatial damage distribution.

Furthermore, the classification obtained by the moment tensor analysis showed a demonstrated

consistency when compared with the classification by the AE standard parameters.

Also the damage orientation obtained by the same methodology showed to be similar with the

bar and load orientation. However, Suaris and van Mier [30] point out that the eigenvalues

decomposition method to explicitly determine the crack orientation is only valid for cases of

traction and pure shear.

This study has shown the potential applications of the truss-like DEM not only to simulate AE

monitoring analysis, but also to provide a better understanding of the relationships between

the basic AE parameters.

Acknowledgements

The authors wish to thank the National Council for Scientific and Technological Development

(CNPq – Brazil) and Coordination for the Improvement of Higher Education Personnel

(CAPES – Brazil) for funding this research.

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