Particle-scale origin of quadrupolar non-affine displacement fields in granular solids

TL;DR

The paper investigates the particle-scale origin of quadrupolar non-affine displacement fields in 2D jammed disk packings subjected to athermal quasistatic simple shear. It develops a discrete Eshelby-equivalent inclusion framework by treating Delaunay triangles as local inclusions with stiffness mismatches and reconstructs the non-affine field as a superposition of triangle eigenstrains applied to a reference network. The study shows that isolated quadrupoles appear with increasing pressure when missing contacts are few and aligned with low-frequency vibrational modes, and that healing nearby missing contacts dissolves these quadrupoles, indicating a structural-defect mechanism. These results provide a particle-scale explanation for deformation in amorphous granular solids and offer a pathway to extend Eshelby-type analyses to discrete systems and higher dimensions, with implications for understanding shear localization and failure in disordered materials.

Abstract

In this work, we identify the local structural defects that control the non-affine displacement fields in jammed disk packings subjected to athermal, quasistatic (AQS) simple shear. While complex non-affine displacement fields typically occur during simple shear, isolated effective quadrupoles are also observed and their probability increases with increasing pressure. We show that the emergence of an isolated effective quadrupole requires the breaking of an interparticle contact that is aligned with low-frequency, spatially extended vibrational modes. Since the Eshelby inhomogeneity problem gives rise to quadrupolar displacement fields in continuum materials, we reformulate and implement Eshelby's equivalent inclusion method (EIM) for jammed disk packings. Using EIM, we show that we can reconstruct the non-affine displacement fields for jammed disk packings in response to applied shear as a sum of discrete Eshelby-like defects that are caused by mismatches in the local stiffnesses of triangles formed from Delaunay triangulation of the disk centers.

Figures (21)

• Figure 1: (a) Shear stress $\Sigma$ plotted versus shear strain $\gamma$ for a bidisperse mixture (diameter ratio $\sigma_l/\sigma_s=1.4$) of $N = 2048$ disks at an initial pressure of $p = 0.1$ ($\phi=1.0$) undergoing athermal, quasistatic simple shear with strain increment $\Delta \gamma = 2 \times 10^{-5}$. The data is shaded from violet to dark red as the fraction of missing contacts, $N_m/(N+1)$, increases. The black arrows in panels (b) and (c) show the corresponding non-affine displacement fields, $\vec{u}_b$ and $\vec{u}_c$, at the strains marked by the circle and $\times$, respectively, in (a). The $\times$ and $+$ signs in panel (a) indicate the strain of the non-affine displacement fields in Fig. \ref{['fig:fit']} (a) and (b). $\vec{u}_b$ and ${\vec{u}}_c$ are magnified by a factor of 40000 and 5000, respectively, to improve visualization.
• Figure 2: Non-affine displacement fields (orange-filled arrows) from the simulations of athermal, quasistatic simple shear in Fig. \ref{['fig:ss']}: (a) at $\gamma=0.057$ fit to a single effective quadrupole with $R^2(1) = 0.87$ and (b) at $\gamma=0.039$ fit to two effective quadrupolar structures with $R^2(2) = 0.79$. The fitted quadrupolar non-affine displacement fields are represented by blue-filled arrows, with their centers marked by filled green circles. When the simulation and fitted non-affine displacement fields overlap, the arrows are shaed black. In (b), the quadrupole on the right (darker green) and the one on the left (lighter green) contribute $55\%$ and $45\%$ to the total non-affine displacement field, respectively. The dashed lines in both panels define the distance $d_{ij}^q$ from the center of quadrupole $q$ to the center of the bond between disks $i$ and $j$.
• Figure 3: (a) Probability that the fitted non-affine displacement field ${\vec{u}}$ possesses $R^2(n_{\rm eff}) > 0.7$ plotted as a function of pressure $p$. The data is obtained by fitting ${\vec{u}}$ at the beginning of each quasi-elastic segment of shear stress versus strain to the non-affine displacement field generated by either one ($n_{\rm eff} = 1$, circles) or two ($n_{\rm eff} = 2$, squares) effective quadrupoles. (See Eq. \ref{['fiteq']}.) The average fractional number of missing contacts $\langle N_m/(N+1)\rangle$ at each $p$ is displayed on the top axis. (b) $\langle N_m/(N+1)\rangle$ plotted as a function of $p$ for the same data in (a).
• Figure 4: Non-affine displacement field ${\vec{u}}$ (black arrows) in response to a single simple shear step $\Delta \gamma=10^{-5}$ (followed by energy minimization) applied to jammed disk packings: (a) A nearly crystalline packing at pressure $p=1.8\times10^{-3}$, with a polydispersity of $\Delta \sigma/\langle \sigma\rangle = 6 \times 10^{-4}$ in diameter and a packing fraction of $\phi=0.9085$; and (b) A disordered bidisperse disk packing at $p=0.285$ and $\phi =1.2$. In both cases, the system contains $N_c = 3N-1$ interparticle contacts, i.e., one missing contact (highlighted by a red thick line) compared to the fully connected Delaunay network. The angle between the missing contact and the shear direction is (a) $\alpha = 2\pi/3$ and (b) $0.4\pi$.
• Figure 5: Magnitude of the non-affine displacement field $\|\vec{u}\|$, normalized by the simple shear strain increment $\Delta\gamma$, for jammed disk packings with one missing contact (relative to the Delaunay-triangulated network) plotted as a function of the angle $\alpha$ of the missing bond relative to the shear direction. The circles are colored by $R^2(1)$ (increasing from violet to dark red) from fits of ${\vec{u}}$ to the displacement field for a single effective quadrupole (Eq. \ref{['fiteq']}). The squares indicate the average $\langle \|\vec{u}\|\rangle /\Delta \gamma$ at each $\alpha$.