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Grain-scale experimental investigation of localised deformation in sand: a discrete particle tracking approach

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Abstract

Recent developments in the application of x-ray micro-tomography in laboratory geomechanics have allowed all the individual grains of sand in a test sample to be seen and identified uniquely in 3D. Combining such imaging capabilities with experiments carried out “in situ” within an imaging set-up has led to the possibility of directly observing the mechanisms of deformation as they happen. The challenge has thus become extracting pertinent, quantified information from these rich time-lapse 3D images to elucidate the mechanics at play. This paper presents a new approach (ID-Track) for the quantification of individual grain kinematics (displacements and rotations) of large quantities of sand grains (tens of thousands) in a test sample undergoing loading. With ID-Track, grains are tracked between images based on some geometrical feature(s) that allow their unique identification and matching between images. This differs from Digital Image Correlation (DIC), which makes measurements by recognising patterns between images. Since ID-Track does not use the image of a grain for tracking, it is significantly faster than DIC. The technique is detailed in the paper, and is shown to be fast and simple, giving good measurements of displacements, but suffering in the measurement of rotations when compared with Discrete DIC. Subsequently, results are presented from successful applications of ID-track to triaxial tests on two quite different sands: the angular Hostun sand and the rounded Caicos Ooids. This reveals details on the performance of the technique for different grain shapes and insight into the differences in the grain-scale mechanisms occurring in these two sands as they exhibit strain localisation under triaxial loading.

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Notes

  1. This is because the centre of mass of a line (beginning pixel a and ending pixel b) takes the following form:

    Defining the length of the line:

    \( {\textbf{\textit{l}}} = {\textbf{\textit{b - a}}} + 1 \)

    \( {\text{COM}} = 1/l \times \sum\limits_{x = a}^b \left( { {\text{x}} + 0.5 } \right) \)

    Making a sum over the length we now get:

    \( {\text{COM}} = {a} + 1/{{l}} \times \sum\limits_{x = 0}^{l-1} \left( { {\text{x}} + 0.5 } \right) \)

    \( {\text{COM}} = {a} + 0.5 + 1/{l} \times \sum\limits_{x = 1}^{l-1} \left( { x } \right) \)

    Since the sum of an arithmetic progression of l numbers starting with a, ending with b, and with difference between successive numbers equal to 1 is given by:

    \( {\text{Sum}} = {\text{l}}/2\left( {\textbf{\textit{a + b}}} \right) \)

    Substituting:

    \( {\text{COM}} = {\textbf{\textit{a}}} + \left( {1/{\textbf{\textit{l}}} } \right) \times {\text{l}}/2 \, \left( {\left( {{\textbf{\textit{a}}} + 0.5} \right) + \left( {{\textbf{\textit{b}}} + 0.5} \right)} \right) \)

    Simplifying:

    \( {\text{COM}} = {\textbf{\textit{a}}} + \left( {1/2} \right)\left( {{\textbf{\textit{a}}} + {\textbf{\textit{b}}} + 1} \right) \)

    Thus the value of COM depends only on a and b—which can only take integer values, and so the COM has a sensitivity (minimum movement) of 0.5 pixels.

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Authors and Affiliations

  1. Grenoble-INP/UJF-Grenoble 1/CNRS UMR 5521, Laboratoire 3SR, 38041, Grenoble, France

    Edward Andò, Gioacchino Viggiani, Jacques Desrues & Pierre Bésuelle

  2. Division of Solid Mechanics, Lund University, Lund, Sweden and European Spallation Source AB, Lund, Sweden

    Stephen A. Hall

Authors
  1. Edward Andò
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  2. Stephen A. Hall
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  3. Gioacchino Viggiani
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  4. Jacques Desrues
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  5. Pierre Bésuelle
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Correspondence to Edward Andò.

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Andò, E., Hall, S.A., Viggiani, G. et al. Grain-scale experimental investigation of localised deformation in sand: a discrete particle tracking approach. Acta Geotech. 7, 1–13 (2012). https://doi.org/10.1007/s11440-011-0151-6

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