DOI: 10.5061/dryad.nzs7h44wt
title: Effect of the loading condition on the statistics of crackling noise accompanying the failure of porous rocks

description: 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.
description: 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/
description: 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
description: # 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
 E<sub>b</sub>. The arrows indicate the strain values where
 cracking starts ε<sub>min</sub>, where final breakdown occurs
 σ<sub>c</sub>, furthermore, the Yield stress
 σ<sub>c</sub>, 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 E<sub>b</sub>. The probability
 distributions p(ε<sub>min</sub>) and
 p(ε<sub>c</sub>) 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
 σ<sub>Y</sub> 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
 Δ<sub>i</sub> of the avalanches *i=1,...,
 n<sub>b</sub>*. The continuous red line indicates the moving
 average of the avalanche size &lt;Δ&gt; 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<sup>-τ</sup>exp(-x/x<sub>0</sub>). The
 duration T and the energy E are made dimensionless by division with the
 correlation time t<sub>c</sub> (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
 ν<sub>E</sub>=1.01 and ν<sub>T</sub>=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
 ν<sub>W</sub>=1.02. **Figure 6.** Probability distribution of
 waiting times p(t<sub>W</sub>) between consecutive avalanches.
 The red line represents the best fit obtained with
 p(t<sub>W</sub>)\~x<sup>-τ</sup><sub><sup>W</sup></sub>exp(-t<sub>W</sub>/t<sub>W0</sub>). **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<sub><sup>\*</sup></sub> 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 &lt;Δ&gt; as function of ε. The vertical dashed line indicates the average value of the strain ε<sub>min</sub>, 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 n<sub>t</sub> and shear n<sub>s</sub>  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 n<sub>t</sub>+n<sub>s </sub>=1 holds for any ε.