Linking initial microstructure and local response during quasistatic granular compaction

We performed experiments combining three-dimensional x-ray diffraction and x-ray computed tomography to explore the relationship between microstructure and local force and strain during quasistatic granular compaction. We found that initial void space around a grain and contact coordination number before compaction can be used to predict regions vulnerable to above-average local force and strain at later stages of compaction. We also found correlations between void space around a grain and coordination number, and between grain stress and maximum interparticle force, at all stages of compaction. Finally, we observed grains that fracture to have an above-average initial local void space and a below-average initial coordination number. Our findings provide (1) a detailed description of microstructure evolution during quasistatic granular compaction, (2) an approach for identifying regions vulnerable to large values of strain and interparticle force, and (3) methods for identifying regions of a material with large interparticle forces and coordination numbers from measurements of grain stress and local porosity.

Figure 1

(a) Schematic of the experimental setup, not to scale. (b) Rendering of granular sample at load step 0 from segmented XRCT image. (c) Average grain stress as a function of height in the sample at five load steps. (d) Load cell versus sample strain. (e) Comparison of load cell and volume-averaged sample stress computed from ff-HEDM data.

Figure 2

(a) All image voxels belonging to a list of potential contact planes. (b) Coordination number as a function of height in the sample at five load steps. (c) Evolution of average coordination number of all grains in the sample. (d) Probability density of coordination number at five load steps. The letter S in the legend indicates load step. (e) Same as (d) but with small and large grain size populations separated. (f) Standard deviation of coordination number as a function of load step and grain size population. (d) and (e) share $x$ axes.

Figure 3

(a) Rendering of the radical Voronoi tessellation for the sample at load step 0. Yellow spheres represent grains and the edges of the tessellation are rendered as blue lines. (b) Porosity as a function of height in the sample at five load steps. (c) Evolution of sample porosity with load step. (d) Probability density of porosity at five load steps. (e) Same as (d) but with small and large grain size populations separated. (f) Standard deviation of porosity as a function of load step and grain size population. (d) and (e) share $x$ axes.

Figure 4

${\epsilon }_{zz}$ component of equivalent continuum strain, colored on a cutaway of the Delaunay triangulation for the sample at five load steps. Only tetrahedrons with nodes having physical location $y<0$ are rendered in order to illustrate the interior of the sample.

Figure 5

(a) Comparison between volume-averaged equivalent continuum strain, ${\overline{\epsilon }}_{zz}$ (EC), and the sample strain (with legend entry XRD CoM) showing a close agreement of the two strain measures. (b) Probability density of ${\epsilon }_{zz}$ at five load steps. (c) Standard deviation of ${\epsilon }_{zz}$ as a function of load step.

Figure 6

Interparticle forces computed from measurements using techniques from Refs. [20, 21, 22] at five load steps. Black lines represent force vectors and are centered at contact points. These lines are scaled linearly in thickness and length according to force magnitude, using the same scale for all load steps. Only forces having normal magnitude $f_n\ge 2\langle f_n\rangle$ are rendered for clarity. Grains corresponding to these forces and having ff-HEDM completeness scores above the retention threshold are colored according to their principal stresses. Red dashed lines highlight force chains that are present in all future load steps. Many force chains visible in early load steps are not visible in later load steps.

Figure 7

(a) Probability density of normal force magnitudes demonstrate a roughly exponential decay above the mean at all load steps. (b) Probability density of mean grain pressures demonstrate a roughly exponential decay above the mean at all load steps.

Figure 8

Mean grain pressure computed for grains having a ff-HEDM completeness score above the retention threshold of 60%, shown for five load steps. Only grains having physical location $y<0$ are rendered for clarity.

Figure 9

Relationship between mean grain pressure, $P_i$, and the maximum force on a grain at load steps 1 (a), 5 (b), and 9 (c). Colors reflect the number of grains per square with the corresponding bivariate state. White horizontal dashed lines indicate the mean and two times the mean force at each load step. The black dashed line is a visual guide for correlation between variables, but not an actual fit to the data. Forces include those at both grain-grain and grain-boundary contacts.

Figure 10

Relationship between grain porosity, computed from the radical Voronoi tessellation, and grain coordination number, $Z$, at load steps 0 (a), 5 (b), and 9 (c). Colors reflect the number of grains per square with the corresponding bivariate state. The white vertical dashed line indicates the mean porosity in all cells of the radical Voronoi tessellation at each load step.

Figure 11

(a) Correlation coefficient between $P_i$ and $\max \left(f_n\right)$ at each load step. (b) Same as (a) but for ${\varphi }_i$ and $Z$. Panels (c) and (d) are the same as (a) and (b), respectively, but show curves for the separate grain size populations. Panels (a) and (c), and (b) and (d), share $x$ axes.

Figure 12

Relationship between grain porosity, ${\varphi }_i$, at load step 0 and response variables (a) maximum interparticle force, (b) mean grain pressure, and (c) grain-centered equivalent continuum strain, all at load step 9. Colors reflect the number of grains per square with the corresponding bivariate state. White dashed lines indicate mean values.

Figure 13

(a) Probability that the maximum force on a grain will exceed the maximum force in the sample at five load steps, as a function of a grain's porosity at load step 0. (b) Probability that the ${\epsilon }_{zz}$ component of grain-centered strain will exceed (in magnitude) the sample strain at five load steps, as a function of a grain's porosity at load step 0. (c) Probability that the mean grain pressure, $P_i$, will exceed the mean grain pressure in the sample at five load steps, as a function of a grain's porosity at load step 0. The horizontal dashed lines represent means computed for all grains in the sample at each load step. Symbols on the ends of the dashed lines correspond to legend entries indicating the corresponding load step. Lines without symbols correspond to all load steps without symbols.

Figure 14

Correlation coefficients between initial porosity and response at all load steps for (a) $\max (f_n$), (b) ${\epsilon }_{zz}^{\left(i\right)}$ (EC), and (c) $P_i$.

Figure 15

Relationship between grain coordination number, $Z$, at load step 0 and response variables (a) maximum interparticle force, (b) mean grain pressure, (c) grain-centered equivalent continuum strain, all at load step 9. Colors reflect the number of grains per square with the corresponding bivariate state. White dashed lines indicate mean values throughout the sample.

Figure 16

(a) Probability that the maximum force on a grain will exceed the maximum force in the sample at five load steps, as a function of a grain's coordination number at load step 0. (b) Probability that the ${\epsilon }_{zz}$ component of grain-centered strain will be below (more compressive than) the mean sample strain at five load steps, as a function of a grain's coordination number at load step 0. (c) Probability that the mean grain pressure, $P_i$, will exceed the mean grain pressure in the sample at five load steps, as a function of a grain's coordination number at load step 0. The horizontal dashed lines represent means computed for all grains in the sample at each load step. Symbols on the ends of the dashed lines correspond to legend entries indicating the corresponding load step. Lines without symbols correspond to all load steps without symbols.

Figure 17

Correlation coefficients between initial coordination number and response at all load steps for (a) $\max \left(f_n\right)$, (b) ${\epsilon }_{zz}^{\left(i\right)}$ (EC), and (c) $P_i$.

Figure 18

(a) Grain porosity versus coordination number for all grains at load step 0. Symbols indicated in the legend represent grains that experience fracture rendering them non-load-bearing (NLB) or load-bearing (LB), as determined from visual analysis of unsegmented XRCT images. (b) Probability that a grain fractures at some stage of compaction as a function of grain porosity at load step 0. (c) Probability that a grain fractures at some stage of compaction as a function of coordination number at load step 0. Dashed lines in all figures indicate mean values throughout the sample.