·Table of Contents
·Methods and Instrumentation
Poisson Temporal Processes in Acoustic Emission Tests
M. Armeite, M. I. López Pumarega,
ENDE, Centro Atómico Constituyentes, COMISIÓN NACIONAL DE ENERGÍA ATÓMICA, Av. Gral. Paz 1499,
1650 San Martín, Pcia. Buenos Aires, ARGENTINA.
Member of GLEA (Latin-American Acoustic Emission Group)
E-mails: firstname.lastname@example.org, email@example.com, firstname.lastname@example.org
Facultad Ciencias Exactas y Naturales, Dpto. Matemática, Universidad de Buenos Aires, ARGENTINA.
ENDE, Centro Atómico Constituyentes, COMISIÓN NACIONAL DE ENERGÍA ATÓMICA, Av. Gral. Paz 1499, 1650 San Martín, Pcia. Buenos Aires, ARGENTINA.
Member of GLEA (Latin-American Acoustic Emission Group)
E-mails: email@example.com firstname.lastname@example.org email@example.com
Grupo Análisis de Señales, Facultad Regional Buenos Aires, Universidad Tecnológica Nacional, ARGENTINA.
The present paper follows previous results, obtained in tests of deformation and crack propagation in samples from stainless steel seamless tubes, concerning the Poisson temporal distribution of Acoustic Emission (AE) signals. A new set of 11 tests was performed, in order to corroborate if they obeyed the same temporal statistics. The present work consisted in analyzing in each test the distribution of the temporal gap between two successive AE events. From the results obtained, hypotheses about the involved physical phenomena related with material deformation and crack propagation are discussed. The influence of the precision of the AE system on results is also studied.
In previous papers, the statistics followed by the parameters Amplitude, Duration and Rise-time, from Acoustic Emission (AE) signals was studied. The signals were bursts or events generated during the deformation, at constant rate, of ring shaped samples coming from stainless steel seamless tubes [1, 2]. A mathematical model that simulated the AE events and fulfilled the same statistics as the experimental data was constructed . The AE signal was considered as the response of a linear system to a train of impulses. The signal could then be represented by a stochastic sequence of bursts or events randomly distributed in time. In order to verify the model a particular distribution of impulses was selected to enter the system. The results from a unique test suggested that the signals followed the Poisson temporal statistics.
In order to verify the preliminary results a set of 12 new similar tests was designed with the aim of corroborating the Poisson dependence, in accordance with the scarce literature available.
1.1. Temporal processes
When the important variable is the number of events, it implies an underlying counting process, as happens in the radioactive nuclear disintegration, the number of phone calls to a phone central office, the number of defective products in a manufacturing process, the number of arrived cosmic rays, etc.
The usual way to describe the occurrence pattern is in terms of temporal gap between two consecutive events. If the arrival pattern does not present variability, the interval between events is constant. But if the arrival of events varies stochastically, it is necessary to define the probability function of the temporal gap between two consecutive events.
The Poisson distribution is widely applied in counting processes. The proper use of this model in the present investigation is guaranteed if the following three conditions hold:
Then, the probability of occurrence of k events in the time interval of fixed length t is given by the function:
- The expected rate of events occurrence is a constant, l, which is the mean number of events in the unit time.
- The probability of occurrence of just one event in the time interval (t, t + Dt) is, in firs approximation equal to lDt.
- If Dt is small enough, the probability of occurrence of more than one event in Dt, tends to zero more rapidly than Dt.
This is a discrete distribution and it is precisely the Poisson distribution. A pattern of stochastic arrivals that follows this distribution is called Poisson arrival pattern. A Poisson process is a stochastic process that describes the location of points (events) in a segment that represents a certain time interval, according to the probabilistic Poisson law .
When the arrival pattern is Poisson-type it verifies that the time interval between two consecutive arrivals is exponentially distributed. This is a continuous distribution with a probability density given by:
A necessary and sufficient characterisation of a Poisson process  is that the probability distribution of the temporal distance between two consecutive events follows the negative exponential of expression (2). This is equivalent to set that:
In a Poisson-type process, the points on the temporal axis are located in a non-ordered way, the lack of order is as high as possible. This maximum disorder degree is in the frame of the Information Theory. It has been demonstrated that among the homogeneous punctual processes with a given density l , Poisson-type processes are those with highest entropy.
In reference , it was reported that for the case of an Al sample, the temporal distribution of AE signals fitted a Polya function distribution. In this case, the probability of occurrence of k events in the observation interval t is expressed by:
This distribution function is characterized by two parameters, the mean number of events in the unit time (l ) and a constant d that is the named "infection probability". This function applies to those cases where there exists dependence between successive events. This means that the previous history influences the present and future facts. It is mathematically demonstrated that if the infection probability tend to zero, then the Polya distribution transforms in Poisson distribution.
1.2. AE tests
In the experiments micro-alloyed steel samples from seamless tubes were tested. Samples were classified in five categories according to material composition and previous condition (1). Table I indicates the main characteristics of the tests performed in the present investigation.
||With notch and fatigue crack
||With thin zone
||With thin zone and oxide
||With notch and oxide
|Table I: Characteristics of tests.
In references  and , the total number of events considered for each test was lower than in the present paper. In our present study, all events recorded by the AE AEDOS system are considered, not taking into account if they were saturated or not. The threshold for detection was set at 500 mV, that is to say, AE signals with amplitude lower than this value were not recorded. The same happened with signals that could not be resolved or those that were superposed.
The necessary and sufficient condition for Poisson-type processes, given by expression  was taken into account. The value of l
was calculated considering the duration of each test and the total number of recorded events.
Fig 1: Distance between consecutive events.
Fig 2: Distance between consecutive events.
In this way, the separation time for consecutive events was studied for the 12 experiments. The respective histograms were obtained with the software STATISTICA 4.5 for Windows. Figures 1 and 2 show examples of the obtained graphs. The analysis of goodness of fit of histograms to the proposed exponential curves was performed with the Kolmogorov-Smirnov test, studying if the p value calculated with the software showed the indication for non refusal of the hypothesis of fitness of the experimental curve to the theoretical curve, for a significance level of 0.05. Table II summarizes the results for each test.
It has to be noted the low variation of the de l
values for each individual test-type, especially for tests type a. On the other hand the variation among tests of different type is important.
|Table II: Characteristics and Poisson parameters for each test.|
After this first analysis, it was concluded that the fitting of the experimental histograms with exponential functions could not be rejected, so the hypothesis that tests obeyed of a Poisson-type process could not be rejected.
In this case the Dead Time was 64000 ms. Another important factor to be considered is the piezoelectric sensor Decay Time. For the broad band sensor employed in our experiments it was in the order of 20 ms , so, being lower than the Dead Time it was not a defining factor for the detection of signals.
Although the AEDOS system uses a precision of 1 ms for the determination of the Duration and Rise-time of events, in the action of recording the arrival time for each event the precision falls down to 1 s. This implies that real separate events could be merged at the same temporal interval. Clearly, this precision in the acquisition time of signals has to be considered as the lowest temporal limit.
Some very significant physical hypotheses are underlying the theory of Poisson-type processes. The experimental conditions, the implied forces and influences that rule the process hold invariable during the time interval t, therefore the probability of occurrence of each individual event is the same in the whole interval. This probability is independent of the previous development of the process and only depends on the present state, that is to say, the time intervals are stochastically independent. The information concerning the number of events corresponding to a given time interval does not reveal anything about other intervals .
The AE phenomenon is an ideal candidate for the application of the Self Organized Criticality (SOC) theory. Taking into account the different mechanisms producing AE that are found in the literature, it is almost impossible to consider that the different events are produced one independent from the other. This means that the AE sources do not act independently. It is more plausible to think of a precursor that induces other emissions in near points at favorable positions, forming the whole AE event. This focus places the AE in the frame of the SOC theory. There are some antecedents in this sense. James and Carpenter referring to dislocations movement, set that if stress is increased, it is feasible that various dislocations become disentangled producing an elastic wave that by the way favor the disentangling of new ones. An avalanche is then produced in a small volume. More recently, Meisel and Cote studied the Barkhausen effect, finding SOC type characteristics in the abundant AE that is typical in this effect. In 1994, Cannelli, Cantelli and Cordero associated the AE generated during the fracture due to hydrogen precipitation in niobium with a SOC type phenomenon. The inclusion of AE as a SOC phenomenon was also considered in another work of our group, where AE signals obtained from the breakdown of a TiN coating of a stainless steel sample were analyzed with wavelets as 1/f processes. All this induces the idea of interrelated events.
Some other considerations can be made, referring to he time intervals of the physical processes involved in the generation of AE in metals. N Natsik y Chishko[14, 15] performed a theoretical estimation on the time employed by a Frank-Read source for the production of a dislocation loop, obtaining values among 0.06 ms and 50 ms, depending on different conditions for the applied load and initial loop length. Imanaka, Sano and Shimizu estimated the time involved in the production of the same sources as 0.01 ms. It is plausible to consider each grain in steel (diameter 30 mm) as the zone where replicas are produced, therefore, the minimum time for obtaining replicas would be of the order of 0.01 ms, for a propagation speed of 3000 m/s.
Up to now we have only considered AE sources and their replicas. Taking into account the geometry of samples, it is also necessary to consider the reflections in each of the free surfaces of the samples. According to their dimensions, the involved times in our experiments are greater than 6 ms.
As we can see, the times involved in the AE production are lower than or equal to the decay time of the sensor, and this is an important difficulty in the discrimination of the signals. From our analysis it can ver inferred that with the employed AE system it is not possible to discriminate individual signals so the Poisson distribution is obtained as a limit case for a Polya distribution with a null infection probability.
We studied and analyzed, in 12 tests, the temporal evolution of AE signals obtained during the deformation and crack propagation in samples coming form steel seamless tubes
It is suggested that the AE in metals is a SOC type process and so it is impossible that the sources act independently. So, the temporal phenomenon would correspond to a Polya type process with interrelated events, which are produced during very short times. This would conduct to the recording of a unique event, involving the original event and its replicas. It is then possible that we have been recording the larger AE events, which are also more separated between each other and would respond to the more disordered Poisson distribution. The hypothesis of interrelated events is based on physical facts.
As was put in the discussion, it is fundamental to take into account the inherent limitation of the detection system, from where it is deduced the importance of performing in the future testes with higher precision in the determination of occurrence time of each AE event.
The authors are grateful to Dr. R. Piotrkowski, for draft correction and English translation.
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