|NDT.net - September 2002, Vol. 7 No.09|
Nucleation of a crack is readily detected by acoustic emission (AE) method. One powerful technique for AE waveform analysis is being developed as SiGMA (Simplified Green’s functions for Moment tensor Analysis), as crack kinematics of locations, types and orientations are quantitatively determined. Because these kinematical outcomes are obtained as three-dimensional (3-D) locations and vectors, 3-D visualization is definitely desirable. To this end, the visualization system is developed by using VRML (Virtual Reality Modeling Language). As an application, failure process of a reinforced concrete beam is discussed.
The generalized theory of acoustic emission (AE) has been established on the basis of elastodynamics (Ohtsu and Ono, 1984). Thus, it is clarified that AE waves are elastic waves due to dynamic dislocation in a solid. Theoretical treatment of AE in concrete was studied as elastic waves in a homogeneous medium (Ohtsu, 1982). The results were remarkably successful, whereas concrete is not homogeneous but heterogeneous. This is because elastodynamic properties of material constituents are physically dependent on the relation between the wavelengths and the characteristic dimensions of heterogeneity. In the case that the wavelengths are even larger than the sizes of heterogeneous inclusions, the effect of heterogeneity is inconsequent. This is the case of massive solids such as concrete and rock, if the sizes of specimens are large enough compared with the wavelengths.
Theoretical treatment of AE waves leads to the moment tensor analysis for source kinematics (Kim and Sachse, 1984) and the deconvolution analysis for kinetics (Hsu and Hardy, 1978). In the former paper, only diagonal components of the moment tensor were assumed to characterize cracking mechanisms of glass due to indentation. Mathematically, the presence of tensor components is not actually associated with the type of the crack, but substantially related with the coordinate system. Although the crack orientations are often assumed as parallel to the coordinate system (Saito, Takemoto, Suzuki and Ono, 1998), they are generally inclined to the coordinate system mostly because of the configuration of the specimen. As a result, the presence of all the components is consequent whether the type of the crack is of tensile or of shear.
In order to perform the moment tensor analysis, one powerful technique for AE waveform analysis is being developed as SiGMA (Simplified Green’s functions for Moment tensor Analysis) (Ohtsu, 1991). Crack kinematics on locations, types and orientations are quantitatively determined. Because these kinematical outcomes are obtained as three-dimensional (3-D) locations and vectors, 3-D visualization of results is desirable. To this end, the visualization procedure is developed by using VRML (Virtual Reality Modeling Language).
Mathematically, such equivalent forces as the dipole forces and the double-couple forces correspond to particular components of the stress in Fig. 2. Normal components of the moment tensor are identical to thedipole forces, while couple forces correspond to tangential (shear) components. The concept of the equivalent forces is sometime so misleading that nucleation of a tensile crack is represented by only a pair of dipole forces. In the case of a pure tensile crack, a scalar product lknk =1. Since all diagonal components contain the scalar product as given in eq. 12, the tensile crack should be modeled by three normal components (three pairs of dipole forces). In contrast, couple forces correspond to off-diagonal components in eq. 12. Since the moment tensor is symmetric, double-couple forces are rational.
For an inverse problem of eq. 11, the spatial derivatives of Green's functions are inevitably required. Accordingly, numerical solutions are obtained by FDM (Enoki, Kishi and Kohara, 1986) and by FEM (Hamstad, O’Gallagher and J. Gary, 1999). These solutions, however, need a vector processor for computation, and are not readily applicable to processing a large amount of AE waves. Based on the far-filed term of P wave, a simplified procedure has been developed, which is suitable for a PC-based processor and robust in computation. The procedure is now implemented as a SiGMA (Simplified Green's functions for Moment tensor Analysis) code. By taking into account only P wave motion of the far field (1/R term) of Green’s function in an infinite space, the displacement Ui(x,t) of P wave motion is obtained from eq. 11 as,
Here r is the density of the material and vp is the velocity of P wave. R is the distance between the source y and the observation point x, of which direction cosine is r = (r1, r2, r3).Considering the effect of reflection at the surface and neglecting the source-time function,amplitude A(x) of the first motion is represented,
where Cs is the calibration coefficient including material constants in eq. 13. t is the direction of the sensor sensitivity. Ref(t,r) is the reflection coefficient at the observation location x. In the relative moment tensor analysis (Dahm, 1996), this coefficient is not taken into consideration, because the effect of the sensor locations is compensated. Since the moment tensor is symmetric, the number of independent unknowns Mij to be solved is six. Thus, multi-channel observation of the first motions at more than six channels is required to determine the moment tensor components.
Displaying AE waveform on CRT screen, two parameters of the arrival time (P1) and the amplitude of the first motion (P2) in Fig. 3 are determined. In the location procedure, source location y is determined from the arrival time differences. Then, distance R and its direction vector r are determined. The amplitudes of the first motions at more than 6 channels are substituted into eq. 14, and the components of the moment tensor Mij are determined. Since the SiGMA code requires only relative values of the moment tensor components, the relative calibration of the sensors is sufficient enough. Then, the classification of a crack is performed by the eigenvalue analysis of the moment tensor. Setting the ratio of the maximum shear contribution as X, three eigenvalues for the shear crack become X, 0, -X. Likewise, the ratio of the maximum deviatoric tensile component is set as Y and the isotropic tensile as Z. It is assumed that the principal axes of the shear crack is identical to those of the tensile crack. Then, the eigenvalues of the moment tensor for a general case are represented by the combination of the shear crack and the tensile crack. Because relative values are determined in the SiGMA, three eigenvalues are normalized and decomposed,
where X, Y, and Z denote the shear ratio, the deviatoric tensile ratio, and the isotropic tensile ratio, respectively. In the present SiGMA code, AE sources of which the shear ratios are less than 40% are classified into tensile cracks. The sources of X > 60% are classified into shear cracks. In between 40% and 60%, cracks are referred to as mixed mode. In the eigenvalue analysis, three eigenvectors e1, e2, and e3,
are also determined. Vectors l and n, which are interchangeable, are recovered from eq. 16.
|Fig 3: Detected AE wave and two parameters P1 and P2.|
Visualization procedure is developed by using VRML (Virtual Reality Modeling Language). By applying a conventional SiGMA code, analytical results for one AE event are listed in Table 1. From the top, event number, moment tensor components and location of AE source in the Cartesian coordinates are denoted. Following normalized eigenvalues, three eigenvectors and X (Shear), Y(CLVD) and Z(Mean) ratios are given. Here, deviatoric component Y is referred to as a compensated linear-vector dipole (CLVD) after Knopoff and Randall (1970). At the bottom line, crack motion vector l and crack normal vector n are shown, which are interchangeable.
|Fig 7: Experimental set-up.|
Tensile cracks are generated first at the bottom region as bending cracks. Then, delamination between concrete and reinforcement occurs. Along with this failure, bending cracks grow further. The tips of cracks extend upward, penetrating into the compressive zone of the upper half. The cracks may stop at this stage due to compression, and the beam reaches final failure of diagonal-shear failure or concrete crashing at the upper half.
(a) First stage
(b) Second stage
(c) Third stage
Fig 8: Results of SiGMA analysis on the beginning three stages.
3-D visualization of SiGMA analysis on the beginning three stages is given in Fig. 8. At the first stage, a few tensile cracks (green) and mixed-mode cracks (red) are mostly observed near reinforcement at the central region. Activity of cracking increases at the second stage as the increase of mixed-mode cracks. At this stage, bending cracks are visually observed. At the third stage, AE cluster expands upward, increasing the number of shear cracks (blue).
The latter two stages are shown in Fig. 9. Cluster of AE sources further expands and nucleation of cracks is really mixed up of tensile, mixed-mode, and shear cracks. It is noted that tensile and mixed-mode cracks are intensely observed around reinforcement, while shear cracks are particularly observed at the compressive zone. At the fifth stage, cracks distribute widely, probably corresponding to nucleation of diagonal shear cracks between the loading point and the support.
(a) Fourth stage
(b) Fifth stage
Fig 9: Results of SiGMA analysis on the last two stages
Combining all results analyzed, Fig. 10 is obtained. As shown, actually all figures are movable and rotatable. So, locations and orientations of the source can be visually identified. This is a good point by means of VRML.
(a) All the data plotted.
(b) Visualization from an inclined angle.
Fig 10: Results of all the data analyzed
Nucleation of cracks can be quantitatively analyzed from AE waveforms, applying SiGMA (Simplified Green’s functions for Moment tensor Analysis) code. Crack kinematics on locations, types and orientations are determined three-dimensionally. Because visualization of results is desirable, three-dimensional visualization procedure for SiGMA analysis is developed by using VRML (Virtual Reality Modeling Language). As discussed, failure process of a reinforced concrete beam is successfully visualized and studied
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