Mesoscale and macroscale kinetic energy fluxes from granular fabric evolution

David M. Walker, Antoinette Tordesillas, and Gary Froyland

Phys. Rev. E 89, 032205 – Published 21 March 2014

Abstract

Recent advances in high-resolution measurements means it is now possible to identify and track the local “fabric” or contact topology of individual grains in a deforming sand throughout loading history. These provide compelling impetus to the development of methods for inferring changes in the contact forces and energies at multiple spatiotemporal scales, using information on grain contacts alone. Here we develop a surrogate measure of the fluctuating kinetic energy based on changes in the local contact topology of individual grains. We demonstrate the method for dense granular materials under quasistatic biaxial shear. In these systems, the initially stable and solidlike response eventually gives way to liquidlike behavior and global failure. This crossover in mechanical behavior, akin to a phase transition, is marked by bursts of kinetic energy and frictional dissipation. Mechanisms underlying this release of energy include the buckling of major load-bearing structures known as force chains. These columns of grains represent major repositories for stored strain energy. Stored energy initially accumulates at all of the contacts along the force chain, but is released collectively when the chain overloads and buckles. The exact quantification of the buildup and release of energy in force chains, and the manner in which force chain buckling propagates in the sample (i.e., diffuse and systemwide versus localized into shear bands), requires detailed knowledge of contact forces. To date, however, the forces at grain contacts continue to elude measurement in natural granular materials like sand. Here, using data from computer simulations, we show that a proxy for the fluctuating kinetic energy in dense granular materials can be suitably constructed solely from the evolving properties of the grain's local contact topology. Our approach directly relates the evolution of fabric to energy flux and makes possible research into the propagation of failure from measurements of grain contacts in real granular materials.

Received 4 December 2013

DOI:https://doi.org/10.1103/PhysRevE.89.032205

Authors & Affiliations

David M. Walker and Antoinette Tordesillas^*

Department of Mathematics and Statistics, University of Melbourne, Parkville, VIC 3010, Australia

Gary Froyland

School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia

^*atordesi@unimelb.edu.au

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Issue

Vol. 89, Iss. 3 — March 2014

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Images

Figure 1
Change in local contact and contact force topology of a group of grains from computer simulations of biaxial compression under [(a)–(d)] constant confining pressure and [(e)–(h)] constant volume. Contacts are represented by bonds whose radius and color (dark to light) indicates the magnitude of the contact force (low to high) at the start [(a) and (e)] and end of a strain interval [(b) and (f)]. An increase (decrease) in contact force magnitude typically accompanies a local increase (decrease) in number of contacts or, alternatively, an increase (decrease) locally in lower (higher) length cycles. The change in the grain's contact forces is summarized in the change in the magnitude of its particle load vector magnitude [(c) and (g)]: The higher the change, the lighter the color. Our measure, which is based on contact topology alone, is shown in (d) and (h). Grain positions in (c) and (g) and (d) and (h) are at the end of the strain interval, identical to (b) and (f).
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Figure 2
The macroscopic stress ratio with respect to increasing axial strain for the constant confining pressure test (CP black squares) and the constant volume test (CV blue triangles). Each test fails through strain localization and the formation of a persistent shear band at the time of the drop in stress ratio from peak.
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Figure 3
Quantifying the change in the local cycle topology of a grain across a strain interval $Δ t$ in the CP system. Cycles are colored by their size. At time $t$ the grain is surrounded locally by one 3-cycle, one 6-cycle, and one 7-cycle. This local cycle topology changes to one 5-cycle, one 6-cycle, and one 10-cycle. The cosine distance between cycle vectors formed using these topologies is $d_{C} = 0.667$ .
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Figure 4
Temporal correlation between fluctuating kinetic energy ( $E_{f}$ ) and nonaffine deformation ( $E_{N A}$ ) with our proposed proxies for energy dissipation $E_{C D}$ and $E_{S C D}$ . All energy measures have been rescaled independently to lie in the range $[0, 1]$ for better visualization of the simultaneous spiking. Traces are plotted for the postpeak stress ratio regime in both samples with the upper plots displaying the first half of these regimes and the lower plots the second half for improved clarity. (a) Constant confining pressure test. (b) Constant volume test.
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Figure 5
Constant confining pressure test (a) spatial visualization of $d_{C} (C_{t}^{i}, C_{t + Δ t}^{i})$ over a strain interval corresponding to a drop in stress ratio following peak stress ratio. (b) Across the same interval a plot of the strong force chain network (color 2), the weak network (color 1), and those chains which fail by buckling across the interval (color 3). (c) As in (a) but showing a spatial visualization of $d_{E} (S_{t}^{i}, S_{t + Δ t}^{i})$ . Constant volume test (d) spatial visualization of $d_{C} (C_{t}^{i}, C_{t + Δ t}^{i})$ over a strain interval corresponding to a large drop in stress ratio well into the high-strain regime. (e) Across the same interval a plot of the strong force chain network (color 2), the weak network (color 1), and those chains which fail by buckling across the interval (color 3). (f) As in (d) but showing a spatial visualization of $d_{E} (S_{t}^{i}, S_{t + Δ t}^{i})$ . Values of $d_{C} (C_{t}^{i}, C_{t + Δ t}^{i})$ and $d_{E} (S_{t}^{i}, S_{t + Δ t}^{i})$ in (a), (c), (d), and (f) are scaled to lie in the range [1,64] to better contrast low and high values.
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Figure 6
Constant confining pressure test. (a) The observed grain displacement field over the same interval in (Fig. 5). (b) Spatial visualization of the incremental nonaffine measure. (c) The nonaffine grain displacement field, i.e., the displacement field obtained by subtracting the motion of the boundary walls from the observed displacement field. Constant volume test [(d), (e), and (f) as (a), (b), and (c)]. Values of the incremental nonaffine measure are scaled to lie in the range [1,64] to better observe the contrast of low and high values. The vectors of the displacement field are scaled by a constant factor to better show the regions of activity.
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