Clustering analysis of AE in rock.
A common feature of failure in a quasi-brittle material such as rock is the development of microcracking, which releases energy in the form of elastic waves called acoustic emission (AE). The AE technique can be used to monitor the evolution of damage, through the entire volume, at various stages of loading. The coordinates of an acoustic source can be obtained from the arrival times of the P-wave at the receivers. The locations of AE provide a picture that forms a basis for the justification of mechanical models of damage and failure (Shah and Labuz, 1995; Ono, 2005).
Specimens of rock were tested in a closed-loop, servo-hydraulic load frame and the data from LVDTs, strain gages, and load cells were stored on a computer and time stamped for synchronizing with AE events. Eight piezoceramic AE sensors, connected to preamplifiers with bandpass filters from 0.1 - 1.2 MHz, were bonded to a specimen using a modified alkyl cyanoacrylate adhesive. One sensor was used as a trigger, with a threshold of 7 mV after amplification at 40 dB gain. The AE signals were recorded using four, two-channel high-speed digitizers, at 20 Msamples per second with a 100-[micro]s recording window, and stored for later analysis. A pre-trigger region, set at 50 [micro]s, was included to ensure capture of the first arrival. Various algorithms for analysis of AE (e.g. first arrival detection, locations, source characterization) were developed and used to interpret failure of rock.
The unknowns for a three-dimensional location, assuming that the P-wave velocity is known, are the three AE source coordinates (x, y, z) and the time of source inception (to). However, the quadratic nature of the distance equation requires a minimum of five receivers to remove any ambiguity in optimizing the source location. The optimization is achieved by minimizing the residual [epsilon] in a least-squares sense using the Levenberg-Marquardt technique:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
where [x.sub.i] and [z.sub.i] are the coordinates of the ith sensor; c is the P-wave velocity; and [t.sub.i] is the arrival time at the ith sensor (e.g. Salamon and Wiebol, 1974).
To find the relative arrival time, the waveform was checked to see when the signal passed a designated threshold value, which was determined from the mean amplitude and standard deviation of the noise during the pre-trigger period. The threshold was set at three times the standard deviation from the mean. When the threshold value was exceeded, the point was stored as the trigger point. The data were scanned preceding the trigger point until the point in which the channel's mean value was obtained; this point was then used for the potential arrival of the event. The next 120 points were checked to see if the trigger threshold was crossed three more times during that interval. If the threshold had indeed been exceeded three subsequent times, the point was then stored as the arrival time. About 65% of the total events were located within 5 mm error. The error for each event was calculated as the average difference between the distances from the calculated location and the estimated arrival times.
One way of describing a geometric figure is by its fractal dimension, which can be thought of as the number of variables required to describe a shape. For example, a line has a fractal dimension of one, a plane has a fractal dimension of two, and a volume has a fractal dimension of three (Mandelbrot, 1983). The use of fractal analysis to monitor the evolution of the failure process is reasonable because a fractal dimension of three represents random events occurring in a volume, and a fractal dimension of two signifies that the events have localized on a failure plane (Hirata et al., 1987; Smalley et al., 1987).
The process used to obtain the fractal dimension is the correlation integral, which compares the proximity of events to one another:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
where [N.sub.R](R < r) = the amount of pairs (pi, pj) with a distance smaller than r (e.g. Knox, 1963; Williams, 1983). The correlation integral can then be used to calculate the fractal dimension (FD):
C (r )= [r.sup.FD] (3)
By varying the value for r, a plot can be made to represent the changing value of the correlation integral versus the distance r. The fractal dimension is the slope of the function on a log-log plot, as shown in Fig. 1.
To demonstrate the clustering analysis, three sets of 100 data points, which can be thought of as AE locations, were placed randomly (i) in a volume (Fig. 1a), (ii) on the surface of a sphere (Fig. 1b), and (iii) along a line (Fig. 1c). The clustering analysis was performed, and the corresponding fractal dimensions were 3.0, 2.0, and 1.0, respectively. In laboratory tests associated with failure of rock, a common feature is a plane of fracture or shear band, so the fractal dimension offers the possibility to identify localization of deformation.
[FIGURE 1 OMITTED]
Simulated Failure Plane
One hundred events were spread randomly across a 50-mm plane (vertical range = 0) and the fractal dimension was calculated to be 2.0. This was then done again with a different vertical variance to simulate AE location error, the tortuousity of a crack, or the thickness of a shear band. This variance was created by assigning a random vertical component to each point, and calculating the fractal dimension. The vertical variance was increased in 5-mm increments until the vertical variance was equal to that of the width (50 mm) and a cube was formed. The results are shown in Fig. 2.
With the points in a plane, the fractal dimension was 2.0 (Fig. 2a). Figures 2b and 2c represent a vertical range of 5 and 20 mm, respectively, and the fractal dimensions were 2.15 and 2.70. Thus, for a location error of about 5 mm, a fractal dimension of 2.2 signifies that failure has localized along a planar feature. As the vertical range increased to the height of the specimen, 50 mm, the locations were spread throughout the volume and the fractal dimension was 3.0 (Fig. 2d). From 25-45 mm vertical range, the fractal dimension of 2.9 does not change appreciably (Fig. 3).
[FIGURE 2 OMITTED]
[FIGURE 3 OMITTED]
It has been documented that the initiation of tensile failure is associated with a damage zone of concentrated microcracks, and an intrinsic zone of finite width is indentified (Labuz and Biolzi, 1998); a distinct fracture does not form until further deformation into the post-peak regime. For shear failure in compression, a shear band is formed by a region of inelastic processes such as microcracking and porosity increase within a thickness of 5 - 10 grain diameters (Riedel and Labuz, 2007). Thus, for both tensile and shear type failure, a fractal dimension greater than 2.0 may be realized upon localization because of the failure processes.
Three Point Bend Test
The first series of experiments involved initiation and propagation of tensile failure through mode I fracture. Smooth boundary (no notch) specimens of Serena sandstone were loaded in a three-point bend configuration. The beams were cut to a length of 310 mm, a width of 80 mm, and a thickness of 32 mm; the distance between supports was 300 mm. A strain-gage based displacement transducer with a gage length of 20 mm was placed at the center position to measure the crack mouth opening displacement (CMOD), which was used as the control signal within the closed-loop, servo-hydraulic load frame.
[FIGURE 4 OMITTED]
Figure 4 shows the located events with error <5 mm at different stages of loading, as indicated by the % of peak load called the load ratio. A migration of AE locations with loading was observed and the AE tracked the propagation of the fracture. Because of the absence of a stress concentrator-the boundary of the beam was smooth-only 18 events were located between the 90 - 100% load ratios (Fig. 4a). Nevertheless, a region of localization, which may be related to an intrinsic length of the rock, was evident (Labuz and Biozli, 1998). A fractal dimension of 2.3 was calculated (Fig. 5) for mode I fracture initiation. As CMOD was increased beyond peak load, 152 events were located between the 100 - 90% load ratios (Fig. 4b). The fractal dimension remained at 2.3, even though the post-peak regime is essentially controlled by crack propagation. From the load ratios of 90 - 80%, the located AE events increased to 239 (Fig. 4c), and fractal dimension decreased to 2.2 (Fig. 5). The AE locations showed greater activity near the tip. Between the 80 - 70% load ratios, the located AE events numbered 329 and the length outlined by AE increased (Fig. 4d); the fractal dimension was slightly less than 2.0.
The clustering analysis showed that the fractal dimension started at about 2.3, certainly influenced by the gradient of axial stress due to bending, but localization was identified independently through a speckle interferometry technique. Crack propagation was responsible for the decrease in the fractal dimension in the post-peak regime.
[FIGURE 5 OMITTED]
A plane-strain (biaxial) compression apparatus (Drescher et al. 1990; Riedell and Labuz 2007) was used to examine shear banding in Berea sandstone. The apparatus allows the failure plane to develop and propagate in an unrestricted manner by attaching a low friction linear bearing to the upper platen. Plane-strain deformation is enforced by a thick-walled steel cylinder called the biaxial frame. A significant number of events (778) were located with < 5 mm error at various stages of loading (Fig. 6).
The clustering analysis was applied to the biaxial experiment by considering the AE locations in groups of 70 events (Fig. 7). From load ratios of 95-98% (lateral displacement of 0.085-0.093 mm), 219 events were located and the fractal dimensions were 2.64, 2.86, 2.59. The value of 2.6 was associated with inelastic deformation, but it is interesting to note a slight increase of the fractal dimension to 2.86 at a lateral displacement of 0.088 mm prior to peak load. From 98-100% peak load (lateral displacement of 0.093-0.101 mm), the fractal dimension dropped to 2.49 (Fig. 6b); a loss of homogeneous deformation was also recorded by the lateral LVDTs. In the post-peak region, the fractal dimension remained at 2.49 (Fig. 6c) and decreased to a value of 2.36 (Fig. 6d), before terminating the test. An examination of the specimen after the experiment showed a localization of deformation in the form of a shear band (Riedell and Labuz, 2007).
[FIGURE 6 OMITTED]
[FIGURE 7 OMITTED]
This research was focused on the development of numerical methods to analyze AE data. Specifically, a location algorithm for estimating the source hypocenter was used to map the failure process of rock under tensile and shear type loading. A clustering analysis based on fractal theory provided a quantitative method for describing the transition from random microcracking to localized failure.
The distribution of events within an increment of load has a fractal structure that can be characterized by a fractal dimension, a reduction of which indicates clustering of hypocenters. A parametric study was used to provide a visual interpretation of a fractal dimension between three (random events) and two (localization of events on a plane), while a series of simulated failure planes with different vertical variance was used to test the sensitivity of the fractal dimension. The algorithms developed were used to interpret AE data from various experiments on rock. The results for the beams with a smooth boundary (no notch) showed that near peak load the fractal dimension was about 2.3, and approached 2.0 in the post-peak phase of loading as crack propagation occurred. Confined, plane strain compression tests involving shear-type failure showed that the fractal dimension was approximately 2.5 at peak load, and as the test progressed and shear banding developed, the fractal dimension decreased to around 2.4.
Partial support was provided by NSF Grant Number CMS-0070062 and the GAANN program of the Department of Education.
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N. IVERSON, C.-S. KAO and J. F. LABUZ
Department of Civil Engineering, University of Minnesota, Minneapolis, MN 55455
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|Author:||Iverson, N.; Kao, C.-S.; Labuz, J.F.|
|Publication:||Journal of Acoustic Emission|
|Date:||Jan 1, 2007|
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