Abstract

Abstract In the present paper acoustic emission (AE) source modeling and calculation of elastic wave propagation in concrete is performed by using the elastodynamic finite integration technique (EFIT). It is demonstrated how various AE volume and dislocation sources (e.g. volume expan-sion, tension cracks, shear cracks) can be implemented by using equivalent body forces as ex-pressed by the corresponding moment tensor. We also present first application results by calcu-lating elastic wave propagation due to AE sources in a reinforced concrete specimen. We de-scribe how numerical modeling techniques like EFIT can be used to test and optimize localiza-tion algorithms and moment tensor inversion in heterogeneous media.

1. Introduction

The elastodynamic finite integration technique (EFIT) is a numerical time domain scheme to model elastic wave propagation in isotropic and anisotropic, homogeneous and heterogeneous as well as dissipative and non-dissipative media [1-3]. EFIT uses a velocity-stress formalism on a staggered spatial and temporal grid. It discretizes the first-order governing equations, i.e. the equation of motion and the stress rate equation. While EFIT has been used frequently to model elastic wave propagation caused by impact hammer or piezoelectric transducers coupled to the surface of a specimen [4,5] no application concerning wave propagation due to AE sources has been presented so far. In the following we describe how AE sources can be implemented into the EFIT framework. To simplify matters we focus on the two-dimensional plane strain problem. The extension to three dimensions is straightforward. In 2D the linear equations of motion in their differential form read

	(1)
	(2)

where v_x, v_y are the velocity components, T_xx, T_yy, T_xy denote the stress tensor components, r is the mass density and f_x, f_y are the volume source densities of body forces (N/m³). The linear stress rate equations in their differential form read

	(3)
	(4)
	(5)

where l and µ are the two Lamé coefficients and g_xx, g_yy, g_xy are the volume source densities of stress rates (Pa/s). The five continuum equations given above can be discretized in space and time by approximating partial derivatives by central differences and using appropriate averaging schemes for the material parameters r and µ. Details of the EFIT procedure can be found in [1] and [3] for example. The locations of the 2D-EFIT field components on a staggered grid are shown in Fig. 1.

Fig 1: 2D-EFIT discretization on a staggered Cartesian grid. The velocity components vx, vy, the stress tensor components Txx, Tyy, Txy, as well as the volume source densities of body forces fx, fy and stress rates gxx, gyy, gxy are located at different positions inside the grid cell.

2. Representation of dislocation and volume sources by equivalent body forces

In general acoustic emission sources can be modeled as internal discontinuities caused by dislocation sources (e.g. cracks) or by volume changes (explosions or rapid phase transformations). By introducing the moment tensor M_pq (p,q = x,y,z in 3D) displacement discontinuities can be replaced by equivalent body forces f_p. For that one can introduce a moment tensor density m_pq with

(6)

Here [u_i] denotes a component of the displacement discontinuity [u (x, t)] at position x (across the fault surface S) and time t, n_j is a component of the normal vector to the fault surface, n, and c_ijpq are the components of the fourth-order elasticity tensor [6]. For an effective point source, i.e. wavelength L is large compared to the fault surface S, the moment tensor can now be written as

(7)

and thus m_pq = dM_pq/dS. With Eq. (7) we can express the displacement u(r,t) at position r and time t due to a general displacement discontinuity [u(x, t)] across S as a convolution of the moment tensor M_pq with the partial derivative of the Green's function, _np/x_q ,

(8)

The Green's function G_np describes the displacement u at position r and time t due to a unity force f at position x at time t. More details about the representation of acoustic emission and seismic sources can be found in [6].
According to Eq. (6) displacement discontinuities can simply be described by the components of the moment tensor density, m_pq. The latter in turn can be represented by couples of equivalent body forces whereas the first index p denotes the force direction and the second index q stands for the direction of the moment arm. In Fig. 2 the four components of the two-dimensional plane strain moment tensor are illustrated. The forces are displayed in red, the moment arms in green.

Fig 2: Representation of the four components of the two-dimensional moment tensor density, mpq, by equivalent body forces. The forces are indicated by red arrows, the moment arms by green lines.

3. Acoustic emission source modeling with EFIT

According to Eqs. (1) and (2) EFIT uses the components of the volume source density of body forces and thus various types of AE sources can easily be implemented as shown in Fig. 3. The four columns in Fig. 3 show the vector of displacement discontinuity [u] and the normal vector to the fault surface, n (1^st column), the corresponding moment tensor representation (2nd column), the body force components on the numerical grid needed to implement the source (3^st column), as well as a wave front characteristic representing the absolute value of the particle velocity vector as calculated by the 2D-EFIT code.
As we can see from the 3^st column four grid cells are needed to implement a point source located at the position of the blue circle. In the pictures pressure forces according to the main diagonal elements of the moment tensor are denoted by red arrows while shear forces corresponding to the secondary diagonal of the moment tensor are given in green.

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Fig 3: Various 2D acoustic emission sources and their implementation in EFIT (see the text for a more detailed explanation).

It is important to point out that the different source types can alternatively be implemented using the components of the volume source density of stress rates instead of body forces. For example the volume expansion source shown in the first row of Fig. 3 can also be implemented by using the source terms g_xx and g_yy in Eqs. (3) and (4). The pure shear crack (fault-slip) in the last row of Fig. 3 is equivalent to a volume shear collapse and thus can also be implemented by using the source term g_xy in Eq. (5). The only difference when using stress rate terms instead of body force terms is that the time history of the input pulse has to be adapted. When in the stress rate formalism a function F(t) is used as time history then the time derivative F'(t) must be used in the case of the body force formalism.

The wave front snapshots in the fourth column show the pressure and shear wave characteristic depending on the AE source type. It can be seen that even for the tension and shear crack both wave modes are generated whereas the volume expansion source produces only pressure waves.

4. Acoustic emission analysis in reinforced concrete

Reinforced concrete consists of reinforcement bars and prestressing tendons in addition to the conglomerate of aggregates and pores embedded in a cement matrix. These heterogeneities influence to a high extent the ultrasonic wave propagation by causing a high attenuation due to scattering. Reflections and mode conversions at the specimen boundaries, cracks, aggregates, reinforcement bars and tendons cause a very complex wave field. Often it is not possible to detect other waves than the P-wave, that is why for the quantitative analysis only the arrival times of the P-waves are used. More details about AE analysis of concrete can be found in various papers, e.g. [7-9].

For the numerical EFIT simulations the geometry of an existing prestressed concrete beam was chosen. The specimen was used for fatigue tests and was assigned for AE measurements as well. The goal of the simulations is to lead to a better understanding of the wave propagation process in the specimen and to investigate the influence of prestressing and reinforcement on wave propagation. Furthermore, synthetic data can be used to test and optimize the inversion algorithms for source location and source mechanism (moment tensor inversion).

The cross section in Fig. 4 (on the right) shows four steel reinforcement bars with diameters of 22 mm in the corners and a polyethylene tendon duct with inner diameter of 100 mm and a wall thickness of 3 mm, filled with mortar and 19 steel strands. The concrete has an A-grading curve with a maximum grain size of 16 mm. The porosity is 1 vol.-% with a maximum pore size of 2 mm. The locations of various AE sources are marked by red asterisks in Fig. 4 (on the right). The black rectangles mark the positions of five sensors where the synthetic waveform data was calculated. The picture on the left shows the discrete 2D EFIT model of the specimen with thousands of grains and pores embedded in the cement matrix. Table 1 shows the material parameters that were used in the simulations.

Fig 4: Cross-section of the reinforced concrete specimen with reinforcement bars and tendon duct as used for the AE investigations. The left picture shows the 2D EFIT model. The picture on the right displays the locations of various AE sources (red asterisks) as well as the positions of the sensors (black rectangles).

Figure 4: Cross-section of the reinforced concrete specimen with reinforcement bars and tendon duct as used for the AE investigations. The left picture shows the 2D EFIT model. The picture on the right displays the locations of various AE sources (red asterisks) as well as the positions of the sensors (black rectangles).

Material	cL [m/s]	cS [m/s]	r [kg/m³]
Cement matrix	3950	2250	2050
Gravel & sand aggregates (mean value)	4400	2500	2610
Steel reinforcement	5900	3200	7820
Polyethylene tendon duct	2300	1200	950
Table 1: Material parameters used for the EFIT calculations.

The capability of EFIT for AE source modeling and wave propagation calculations is demonstrated by wave front pictures (Figs. 5 and 6) according to two of the four different source loca-tions represented in Fig. 4 (on the right). One position is inside the cement matrix between the tendon duct and the right boundary of the specimen, another position is directly at the tendon/concrete interface. In order to show the effect of an adjacent boundary we used an isotropic source as described in Fig. 3 (first row). The implementation of other source mechanisms is straightforward as was demonstrated in section 3. The input pulse that was used had a maximum frequency of 250 kHz. The discretized model in Fig. 4 (on the left) consisted of 918 x 918 grid cells (440 x 440 mm², Dx = Dy = 479.8 µm). Moreover 3478 time steps were used (t_max = 200 µs, Dt = 57.5 ns).

Fig 5 (Slide Show): 2D EFIT simulation of elastic wave propagation caused by an isotropic AE source in the cement matrix. The wave front snapshots represent the absolute value of the particle velocity vector using a linear color scale. The wave field is shown in equidistant time intervals of 11.5 µs.

Fig. 5 shows the 2D EFIT simulation of elastic wave propagation caused by an isotropic source in the cement matrix. One can see that this type of source produces only pressure waves and that shear waves are generated later by mode conversion at internal and external boundaries. Additionally it is remarkable that the main parts of the elastic waves are diffracted around the tendon duct and only a small portion of the wave field is transmitted through the duct. This is most likely caused by the large impedance mismatch between concrete and the polyethylene wall (compare acoustic parameters in Tab. 1).

Fig 6 (Slide Show): 2D EFIT simulation of elastic wave propagation caused by an isotropic AE source at the tendon/concrete interface. The wave front snapshots represent the absolute value of the particle velocity vector using a linear color scale. The wave field is shown in equidistant time intervals of 11.5 µs.

Fig. 6 shows a numerical simulation where the isotropic source is located directly at the tendon/concrete interface. This causes the generation of shear waves in addition to the pressure waves. Moreover Lamb waves can be identified within the tendon wall. Similar to the previous case (Fig. 5) the body waves (P and S) are diffracted around the tendon and only a small portion of the waves is transmitted through the duct.

One can see from the wave front pictures that the tendon duct has a significant effect on the wave propagation and thus can influence the accuracy of source localization algorithms. By calculating the time signals of velocity or displacement at the positions of the sensors as exemplary shown in Fig. 7, the numerical simulations can systematically be used to optimize the inversion algorithms, for example by taking into account the location and geometry of the tendon ducts as well as the specific peculiarities of the wave propagation.

Fig 7: Time signals of normal velocity component at five different sensor positions as calculated by the numerical EFIT code. The corresponding wave propagation process is shown in Fig. 5.

5. Conclusions and outlook

The maintenance of existing concrete structures, which plays a more and more important role in structural engineering, requires the development of new non-destructive testing methods. Com-pared to other ultrasonic methods with active sources (pulse-echo or impact-echo method), AE analysis is able to directly monitor the fracture progress, even before cracks become visible on the surface [8]. Other than for metal and fibre reinforced polymers, up to the present there are no established AE methods available for concrete yet. The complex wave propagation in this material shown in the previous simulations makes it difficult to develop generally valid evaluation routines. To find possible applications in the field of condition assessment and monitoring, the localization of the AE sources has to be reliable and a correlation between the signals and the failure processes has to be found. The AE source mechanisms due to crack formation and grow significantly affect the wave-form data as shown in Fig. 3. In principle these crack mechanisms can be inverted from the wave-forms with moment tensor inversion (MTI, e.g. [7,10]). But similar to the localization algorithms the MTI is also based on the assumption of a homogeneous background medium and it is still uncertain how heterogeneities like aggregates, pores, reinforcing bars and tendon ducts affect the validity and accuracy of the inversion algorithms. By using the numerical EFIT code systematic parameter studies can be carried out in order to test and optimize localization and MTI. The important advantage of a numerical method like EFIT compared to experimental measurements is that reproducable AE sources at fixed locations and with clearly defined fracture mechanisms can be implemented. Therefore the numerical calculations are very helpful for validating the inversion methods and for planning experimental test setups.

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