Statistical Properties of Fracture Precursors

We present the data of a mode-I fracture experiment. The samples are broken under imposed pressure. The acoustic emission of microfractures before the breakup of the sample is registered. From the acoustic signals, the position of microfractures and the energy released are calculated. A measure of the clustering of microfractures yields information about the critical load. The statistics from energy measurements strongly suggest that the fracture can be viewed as a critical phenomenon; energy events are distributed in magnitude as a power law, and a critical exponent is found for the energy near fracture. [S0031-9007(97)04346-9] PACS numbers: 62.20.Mk, 46.30.Nz Fracture is a problem which has recently received a lot of attention in the physics community [1–3]. It is troublesome to calculate the force needed to break a heterogeneous material. Instead, it is customary to resort to tests involving the destruction of the sample. Therefore it is interesting to provide additional knowledge about cracks by studying the events that occur prior to the fracture. Besides , despite great experimental and numerical efforts [1– 6], many aspects still remain unclear about the fracture process itself. Conceptually simple models, such as per-colation [6] and self-organized criticality [7], are attractive but often fail to convey the complex phenomenology observed. The main motivation of this work is to understand if these models can reproduce the main features of crack formation. We report here some experimental results that may help to gain valuable information in that direction. Our main tool is the monitoring of the microfractures, which occur before the final breakup, by recording their acoustic emissions (AE). Because of its ability to pinpoint the emission source, this technique has been widely used in seismography and to map the nucleation of fractures [8]. From these signals, we have also obtained the acoustic energy of each microfracture, which is a fraction of the total energy released. The behavior of the energy just before fracture is a good parameter to compare with the above mentioned models. In order to avoid noise, we have designed a setup in which there are no moving parts, the force being exerted by pressurized air (see Fig. 1). A circular sample having a diameter of 22 cm and a thickness of 5 mm is placed between two chambers between which a pressure difference P ෇ P 2 2 P 1 is imposed. The deformation of the plate at the center is bigger than its thickness, then the load is mainly radial [9,10]. Therefore, the experience can be thought of as a mode-I test with circular symmetry. The pressure difference P supported by the sample is slowly increased and it is monitored by a differential transducer. This measure has a stability of 0.002 atm. The fracture pressure for the different tested materials ranges from 0.7 to 2 atm. We regulate P by means of a feedback loop and an electronically controlled valve which connects one of the two chambers to a pressurized air reservoir. The time taken to correct pressure variations (about 0.1 s) is smaller than the characteristic time of the strain rate. An inductive displacement sensor gives the deformation at the center of the plate with a precision of about 10 mm (the deformation just before fracture is of the order of 1 cm). The apparatus is placed inside a copper box covered with a thick foam layer to avoid both electrical and acoustical noise. Four wide-band piezoelectric microphones are placed on the side of the sample (see Fig. 1). The signal is amplified, low-pass filtered at 70 kHz, and sent to a digitizing oscilloscope and to an electronic device which measures the acoustic energy detected by the microphones. The signal captured by the oscilloscope is sent to a computer where a program automatically detects the arrival time of the AE at each microphone. Afterwards, a calculation yields the position of the source inside the sample. A fraction of the detected events is rejected, either as a result of a large uncertainty of the location, or because they are regarded as noise. The mean standard error for the calculated positions is about 6 mm, which results mainly from the uncertainty of the arrival time. The electronic device that measures the energy performs the square of the AE amplitude and then integrates it over a time window of 30 ms, which is the maximum duration of one acoustic event. The output signal is proportional to the energy of the events [11], and