Abstract

In collaboration between the Institute for Computational Physics (ICA1) and the Institute of Construction Materials (IWB) of the University of Stuttgart the evolution of effective force chains percolating through a granular system under uniaxial compression is investigated. It was the idea to prove the theory developed for these granular systems at the ICA1 by a series of experiments using acoustic emission (AE) analysis.

It was predicted, that the percolating effect and the rearrangement of granular materials results in a spontaneous release of acoustic energy radiating waves similar to that observed in other brittle materials under load. After a brief description of the theoretical work done by the group of Hidalgo, Kun and Herrmann this paper focuses to the acoustic emission part of the experiments. The restructuring of spherical beads under compression can clearly be observed by appropriate AE equipment. A correlation of the mechanical test parameters and the results of the AE analysis to computer simulations are giving a better understanding of the behaviour of coarse materials.

Keywords:
Acoustic emission, percolation theory, granular materials

Theoretical Background

Recently, granular materials have been extensively studied under various conditions due to their scientific and technological importance. Huge experimental and theoretical efforts have been devoted to obtain a better understanding of the global behaviour of granular media in terms of microscopic phenomena which occur at the level of discrete particles [Liu & Nagel et al. 1995; Travers & Bideau et al. 1986; Miller & O’Hern et al. 1996; Herrmann & Stauffer et al. 1987; Radjai & Jean et al. 1996; Rintoul & Torquato 1996; Makse & Johnson et al. 2000]. Subjecting a confined granular packing to a uniaxial compression a rather peculiar constitutive behaviour can be observed: for small strains a strong deviation from the linear elastic response can be found implying that the system drastically hardens in this regime [Travers & Bideau et al. 1986; Makse & Johnson et al. 2000]. Linear elastic behaviour can only be achieved asymptotically at larger deformations when the system gets highly compacted. When the external load is decreased again the system shows an irreversible increase in its effective stiffness, furthermore, under cyclic loading hysteretic behaviour is obtained.

Fig 1: Sketch of the used model.

Microscopically, inside a compressed granular packing, stresses are transferred by the contact of particles (Fig. 1). Under gradual loading conditions the particles (with diameter I₀) get slightly displaced changing their contacts and the local load supported by them, which can be experimentally visualized using photoelastic materials for grains. These experiments revealed that in a compressed granular system (with system length h) the stresses are transmitted along the direction of the external load by force chains which can branch at the grains and form a complex network along the specimen [Liu & Nagel et al. 1995]. Particles lying between lines of the force network do not support any load and can even be removed from the packing without changing its mechanical properties. Increasing the external load, more and more force lines appear and they all undergo erratic changes until the system reaches a saturated state when all the particles hold typically the same load and the system behaves as a bulk material. The creation and restructuring of percolating force chains implies relative displacements of particles which can be followed experimentally by recording the acoustic waves emitted, however, up to now no such experiments have been performed systematically. Theoretically, this problem has been mainly studied by means of contact dynamics simulations using spherical or cylindrical particles, and cellular automata [Radjai & Jean et al. 1996; Rintoul & Torquato 1996; Makse & Johnson et al. 2000]. Computer simulations also revealed the generation and evolution of force chains in compressed granular materials. However, the statistics of microscopic restructuring events, the emergence of the array of force chains and their relation to the macroscopic constitutive behaviour remained unclarified. This is why there is such a demand on experimental investigations.

Experimental Setup

We performed experiments by compressing an ensemble of spherical particles in a cylindrical container monitoring the macroscopic constitutive behaviour and the acoustic signals. The cylindrical container was made of PMMA and filled with several thousand spherical beads. It has a thickness of 5 mm and a diameter of 140 mm (Fig. 2). To record the acoustic emission signals, eight broadband sensors sensitive in the frequency range of 50-250 kHz have been glued to the sidewall of the container. The position of the sensors was chosen in a way that subsequently a 3D localization can be carried out using the sensor data of the acoustic emissions. A review of appropriate localization techniques along with some applications can be found in the literature [Grosse 2000].

Fig 2: Details of the PMMA container (dimensions in mm) filled with glass beads.

In a first test series beads out of glass have been used. Additional materials as steel and polymers are discussed; the results of the further experiments will be presented later elsewhere. The data represented in the following section were obtained using either glass beads with I₀ = 5 mm or with I₀ = 8 mm, respectively, to test the theory with coarse material of different diameter.

Usually, the AE signal energy is relatively weak and a proper coupling of the sensors is required. To enhance the data quality in regard to the signal-to-noise ratio the space between the beads was saturated with water. The compressional wave velocity was measured using ultrasound in through-transmission (transmitter – receiver at opposite sides). Besides the material properties, the effective velocity depends on the diameter I₀ of the beads. The standard error of the mean was calculated out of 20 single measurements. For the combination water/glass-beads we found:

I₀ = 5 mm: v_p = 1966 (± 13) m/s
I₀ = 8 mm: v_p = 1430 (± 54) m/s

Replacing the water with glycerine the velocities were shifted to values between 2000 and 2200 m/s. This will give better results for the experiments, where the locations of the AE will be investigated.

AE Experiments – Preliminary Results

A uniaxial pressure system (punch test) was used to apply monotonically increasing displacements at the top level of the glass beads (Fig.3). To reduce the pressure to be obtained by the side wall of the cylinder causing elastic deformations the punch was perforated to let the water flow to both sides of the punch. The actual force and the displacement, measured at the crosshead of the loading machine, together with the AE signals were recorded simultaneously.

Fig 3: Experimental setup showing the cylinder prepared for the punch test and some of the AE sensors.

An eight channel transient recorder was used as an analogue-to-digital converter to enable the storing of the AE waveforms and a signal-based data processing in further experiments. As a first approach to combine the AE data analysis with the simulation in the frame of percolating force chains theory a statistical evaluation of the acoustic emissions was done. Typically several hundred signals were recorded at all channels during the experiments.

To give a preliminary impression of the results, the experimental data of a test using glass beads with I₀ = 5 mm in the water filled container under load are represented in Fig. 4. The experiment was controlled with monotonically increasing displacement. As predicted by the theory multiple rearrangements of the compressed glass beads ensemble occurred, what is indicated by a sudden release of the load. The release of load energy transformed into energy to rearrange the granular material is significantly higher at the end of the experiment than in the beginning. Evaluating the histogram plot in Fig. 4 it is obvious, that this release is accompanied by acoustic emission activities caused by the internally restructuring of percolating force chains. There is a good coincidence between the sharp decline of the load and the occurrence of high energy AE events. The AE data shown were automatically extracted by an algorithm determining the peak amplitudes of the burst signals versus time in form of energy histograms. This energy is defined as the integral of the AE signal amplitude following the onset time. The data recorded at all eight sensors are plotted into this graph to elucidate the time dependent evaluation of AE activity.

Fig 4: Example of the AE measurements and mechanical data. The statistics of AE is plotted using the integral energy at all eight sensors along with the load (straight red line).

However, this experimental approach is straight forward and will be modified for further experiments. There are several points, which need some improvement including a more detailed signal analysis. This is especially true for the discrimination capabilities of signals to noise concerning parameter-based AE approaches. It was shown by Grosse [1996] that a statistical analysis of AE signals without localizing the events can be somehow misleading. Since signal energy seems to be a stable parameter it has to be examined if the energy measured at all sensors is about the same or if mean energy values are representing the acoustic effects in the material more precisely. The statistical analysis procedures should be cross-checked by a signal-based analysis of AE activities including an examination of the influence of preset threshold values.

The configuration of a new multi-channel AE system with high resolution and acceptable recording speed is undergoing. Additional experiments will be conducted using this new AE device investigating glass beads and beads out of other materials.

A more detailed description of acoustic emission data analysis and especially signal-based techniques can be found in Grosse & Reinhardt et al. [1997], Grosse & Weiler et al. [1997] or Grosse & Reinhardt et al. [2002].

Verification and Outlook

Based on the analogy of force lines percolating through the system and fibres of a fibre composite Hidalgo, Kun and Herrmann proposed a novel theoretical approach, namely, an inversion of the Continuous Damage Model (CDM) of fibre bundles [Kun & Zapperi et al. 2000; Hidalgo & Kun et al. 2001] to describe the stress transmission through granular assemblies. The model provides a nonlinear constitutive behaviour in good quantitative agreement with the experimental results as shown in Hidalgo & Grosse et al. 2002. For a system of hard particles the model predicts a universal power law divergence of stress when approaching a critical deformation. The amplitude distribution of acoustic signals was found experimentally (Fig. 5) to follow a power law with exponent d = 1.15 ± 0.05 which is in a good agreement with the analytic solution of the model.

Fig 5: A power law relation was found to give best fitted results according to the size distribution of the signals of the inset [Hidalgo & Grosse et al. 2002].

More results combining simulations, analytical models and experimental data are in preparation to be published soon. This will include new data using different type of material for the beads as well as acoustic emission data with source locations.

Acknowledgement

This work was supported by the Deutsche Forschungsgemeinschaft (DFG) via the collaborative research project "Sonderforschungsbereich" SFB381, and by the NATO grant PST.CLG.977311. F. Kun acknowledges financial support of the Bólyai János Fellowship of the Hungarian Academy of Sciences and of the Research Contract FKFP 0118/2001.

References

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