Effect of the loading condition on the statistics of crackling noise accompanying the failure of porous rocks
Data files
Nov 10, 2023 version files 66.57 MB
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README.md
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repositorium.tgz
Abstract
We test the hypothesis that loading conditions affect the statistical features of crackling noise accompanying the failure of porous rocks by performing discrete element simulations of the tensile failure of numerical rock samples and comparing the results to those of compressive simulations of the same specimens. Cylindrical samples are constructed by sedimenting randomly sized spherical particles connected by beam elements representing the cementation of granules. Under a slowly increasing tensile load, the cohesive contacts between particles break in bursts whose size fluctuates over a broad range. Close to failure breaking avalanches are found to localize on a highly stressed region where the catastrophic avalanche starts and the specimen falls apart into two pieces along a spanning crack. The fracture plane has a random position and orientation falling most likely close to the center of the specimen perpendicular to the load direction. In spite of the strongly different strengths and spatial structure of damage of tensile and compressive failure of numerical rocks, our calculations revealed that the size, energy, and duration of crackling avalanches, and the waiting time between consecutive events all obey scale free statistics with power law exponents which agree within the error bars in the two loading cases.
README: Effect of the loading condition on the statistics of crackling noise accompanying the failure of porous rocks
The primary data files were obtained from a simulation program which generates the breaking process of numerical porous rock samples under slowly increasing elongation. The raw data obtained from discrete element simulations are further processed by a data evaluation program which identifies crackling avalanches, determines their characteristic quantities along with the probability distributions of the quantities of interest. To further analyse the data we used two main freely available open source program packages:
- The GLE Graphics program package was used to carry out the data analysis and the graphical representation of the numerical results. GLE is available under the link: https://glx.sourceforge.io/index.html
- To generate snapshots of the simulations, the povray package was used which is available under the link: http://www.povray.org/
Description of the data and file structure
The data is organized in folders according to the figures of the manuscript. In each folder all the data files of the figure are stored together with the GLE script files (with .gle extension) and with the .c code which was used to generate the data files from the raw data of simulations. Both the raw data of simulations and the processed data files used by the GLE scripts are available in the directories.
Sharing/Access information
Not applicable.
Code/Software
To open a figure in a directory, the .gle script file has to be opened either using the GUI called qgle of GLE or from the command line using the command
gle -d eps filename.gle
This later generates an .eps file of the figure.
If a figure presents snapshot(s) of the simulation, the povray file (with .pov extension) is also available in the folder. To render the image the following command has to be executed
povray +w640 +h480 +x -D -v -ifilename.pov
which renders the file called filename.pov.
For the case of need, the content of the columns of data files are explained in a separate file called repositorium_read_me.doc
Description of the content of the figures whose data are stored in separate directories
Figure 1. (a) Setup of the numerical experiments. DEM simulations were performed by slowly elongating cylindrical samples composed of a random packing of spherical particles. A few particle layers (highlighted in grey) were clamped at the top and bottom of the sample which were then slowly moved against each other along the cylinder axis. (b) Constitutive curve σ(ε) (black) of the system together with the accumulated damage d(ε) (red) in a single sample. The stress is scaled with the Young modulus of beams Eb. The arrows indicate the strain values where cracking starts εmin, where final breakdown occurs σc, furthermore, the Yield stress σc, where the first discernible deviation from linearity of the curve occurs.
Figure 2. Comparison of the constitutive curves σ(ε) of the same sample measured under uniaxial compression (red) and elongation (black). The stress is scaled with the Young modulus of the beams Eb. The probability distributions p(εmin) and p(εc) of the strain of crack initiation (circle) and of global failure (square) are also presented for both loading cases. Filled and open symbols are used for the tensile and compressive results respectively. The dashed lines represent fits of the distributions with the Weibull distribution. The horizontal arrow indicates the Yield stress σY of the compressive case.
Figure 3. Sequence of breaking avalanches of beams along with the constitutive curve σ(ε) of a representative sample. The height of the bars represent the size Δi of the avalanches i=1,..., nb. The continuous red line indicates the moving average of the avalanche size <Δ> obtained by averaging over 51 consecutive events.
Figure 4. Probability distribution of the size p(Δ) (a), duration p(T) (b), and dissipated energy p(E) (c) of bursts taking into account all the events which occurred before catastrophic collapse (up to the maximum of the constitutive curve σ(ε)). The continuous red lines represent fits with the functional form p(x)~x-τexp(-x/x0). The duration T and the energy E are made dimensionless by division with the correlation time tc (the time scale of load redistribution) and the average energy dissipated by a single beam breaking, respectively.
Figure 5. (a) Average energy and duration of avalanches of size Δ. The straight lines represent power laws of exponent νE=1.01 and νT=0.77. (b) Average value of the waiting time elapsed before and after avalanches of size Δ. Power law correlation is revealed between the time elapsed after the avalanche and the size Δ with the correlation exponent νW=1.02.
Figure 6. Probability distribution of waiting times p(tW) between consecutive avalanches. The red line represents the best fit obtained with p(tW)~x-τWexp(-tW/tW0).
Figure 7. Spatial structure of damage in the specimen. (a) Broken beams of avalanches which occurred before the catastrophic event. The circle highlights the region where the last 5 avalanches occurred before the catastrophic one, this is also the region where the catastrophic avalanche was initiated. (b) All the broken beams including also the ones of the catastrophic avalanche. (c) The oblate ellipsoid obtained for the point cloud of the center of the broken beams of the catastrophic event. Avalanches occurred before the catastrophic one are indicated by spheres with a radius proportional to the avalanche size. The color of the spheres indicates the time t when the avalanche occurred according to the color scale on the right, where t is scaled with the critical time t* of failure. Angle Φ enclosed by the shortest axis of the ellipsoid with the load direction is also highlighted.
Figure 8. Average distance of consecutive avalanches together with the average avalanche size <Δ> as function of ε. The vertical dashed line indicates the average value of the strain εmin, where micro-cracking starts.
Figure 9. (a) Probability distribution p(Φ) of the angle Φ between the normal of the plane of the oblate ellipsoid, representing the fracture plane, and the load direction. The value of the average angle is highlighted by the vertical dashed line. (b) Probability distribution p(z) of the position z of the fracture plane along the vertical axis of the cylinder. (c) Probability distribution p(C/A) of the parameter C/A characterizing the sharpness of the fracture plane.
Figure 10. The fraction of beams failing dominantly under tension nt and shear ns are plotted as a function of strain ε for a single sample along with the corresponding constitutive curve for uniaxial tension (a) and compression (b). Note that nt+ns =1 holds for any ε.
Methods
The primary data files were obtained from our simulation program which generates the breaking process of numerical porous rock samples under slowly increasing elongation. The raw data obtained from discrete element simulations are further processed by a data evaluation program which identifies crackling avalanches, determines their characteristic quantities along with the probability distributions of the quantities of interest. To further analyse the data we used two main freely available open source program packages:
- the GLE Graphics program package was used to carry out the data analysis and the graphical representation of the numerical results. GLE is available under the link: https://glx.sourceforge.io/index.html
- To generate snapshots of the simulations, the povray package was used which is available under the link: http://www.povray.org/
Usage notes
Data analysis and graphical representation of the numerical results were carried out using two freely available open source program packages:
- the GLE (Graphics Layout Engine) program package was used to carry data the data analysis and the graphical representation of the numerical results. GLE is available under the link: https://glx.sourceforge.io/index.html
- To generate snapshots of the simulations, the povray package was used which is available under the link: http://www.povray.org/
The data is organized in folders according to the figures of the manuscript. In each folder all the data files of the figure are stored together with the GLE script files (with .gle extension) and with the .c code which was used to generate the data files from the raw data of simulations. The open a figure the .gle script file has to be opened either using the GUI called qgle of GLE or from the command line using the command "gle -d eps filename.gle". The later generates a .eps file of the figure.
If a figure presents snapshot(s) of the simulation, the povray file (with .pov extension) is also available in the folder. To render the image the following command has to be executed
povray +w640 +h480 +x -D -v -ifilename.pov
which renders the file called filename.pov.
For the case of need, the content of the coloumns of data files is explained in a separate file called repositorium_read_me.doc