Stress-strain behavior and geometrical properties of packings of elongated particles

We present a numerical analysis of the effect of particle elongation on the quasistatic behavior of sheared granular media by means of the contact dynamics method. The particle shapes are rounded-cap rectangles characterized by their elongation. The macroscopic and microstructural properties of several packings subjected to biaxial compression are analyzed as a function of particle elongation. We find that the shear strength is an increasing linear function of elongation. Performing an additive decomposition of the stress tensor based on a harmonic approximation of the angular dependence of branch vectors, contact normals, and forces, we show that the increasing mobilization of friction force and the associated anisotropy are key effects of particle elongation. These effects are correlated with partial nematic ordering of the particles which tend to be oriented perpendicular to the major principal stress direction and form side-to-side contacts. However, the force transmission is found to be mainly guided by cap-to-side contacts, which represent the largest fraction of contacts for the most elongated particles. Another interesting finding is that, in contrast to shear strength, the solid fraction first increases with particle elongation but declines as the particles become more elongated. It is also remarkable that the coordination number does not follow this trend so that the packings of more elongated particles are looser but more strongly connected.

Figure 1

Shape of a RCR.

Figure 2

(Color online) Representations of cap-to-cap, cap-to-side, and side-to-side contacts and they will be referred as $cc$ contacts, $cs$ contacts, and $ss$ contacts, respectively.

Figure 3

Boundary conditions for (a) isotropic and (b) biaxial compactions.

Figure 4

Examples of the generated packings at the initial state.

Figure 5

(Color online) Normalized shear stress $q/p$ as a function of cumulative shear strain ${\epsilon }_q$ for different values of the shape parameter $\eta$.

Figure 6

(Color online) Internal angle of friction ${\phi }^{\ast }$ as a function of (a) aspect ratio $\alpha$ and (b) elongation $\eta$. Error bars represent the standard deviation in the residual state.

Figure 7

(Color online) (a) Cumulative volumetric strain ${\epsilon }_p$ as a function of shear strain ${\epsilon }_q$. (b) Solid fraction as a function of particle shape parameter $\eta$ at different levels of shear strain.

Figure 8

(Color online) Dilatancy angle $\psi$ vs internal angle of friction $\phi$ for different values of $\eta$.

Figure 9

Polar representation of the probability density $P_{\vartheta }$ of particle orientations $\vartheta$ for $\eta =0.7$ at various stages of deformation ${\epsilon }_q$. The symbols are the simulation data. Solid lines represent harmonic fit according to Eq. (18).

Figure 10

Particle orientation anisotropy $a_p$ as a function of shape parameter $\eta$ at different stages of shear ${\epsilon }_q$.

Figure 11

(Color online) Color level map of particle orientations for $\eta =0.2$ (up) and $\eta =0.7$ (down) in the residual state.

Figure 12

(Color online) Initial and residual coordination numbers as a function of shape parameter $\eta$. Error bars represent the standard deviation in the residual state.

Figure 13

(Color online) Connectivity diagram for each samples expressing the fraction $P\left(c\right)$ of particles with exactly $c$ contacts in the residual state. Note that the floating particles (i.e., with more than one active) are removed from the statistics

Figure 14

(Color online) Color map of particle connectivities. Color intensity is proportional to coordination number.

Figure 15

(Color online) (a) Contact frame $\left(\mathit{n},\mathit{t}\right)$ and (b) intercenter frame $\left({\mathit{n}}^{\prime },{\mathit{t}}^{\prime }\right)$.

Figure 16

(Color online) Contact anisotropy $a_c$ (black circle) and branch vector anisotropy $a_c^{\prime }$ (red/gray square) as a function of shape parameter $\eta$ averaged in residual state. Error bars represent the standard deviation in the residual state. Inset shows the angular probability densities $P_{\theta }\left(\theta \right)$ in black circle and $P_{{\theta }^{\prime }}\left({\theta }^{\prime }\right)$ in red/gray square for $\eta =0.7$ calculated from the simulation data (points) together with the harmonic approximation (lines).

Figure 17

(Color online) Normal and tangential branch length anisotropies $a_{ln}$ (circle) and $a_{lt}$ (square) and branch length anisotropy $a_{l{n}^{\prime }}$ (triangle) as a function of shape parameter $\eta$ in the residual state. Error bars represent the standard deviation in the residual state. Inset shows the angular average functions $⟨{\ell }_{ln}⟩\left(\theta \right)$, $⟨{\ell }_{lt}⟩\left(\theta \right)$, and $⟨{\ell }_{l{n}^{\prime }}⟩\left(\theta \right)$ in black (circle), red (dark gray square), and green (light gray triangle), respectively, for $\eta =0.7$ calculated from the simulation data (points) and approximated by harmonic fits (lines).

Figure 18

(Color online) Normal and tangential force anisotropies $a_n$ (black circle, plain line) and $a_t$ (black circle, dashed line) and radial and orthoradial force anisotropies $a_n^{\prime }$ (red/gray square, plain line) and $a_t^{\prime }$ (red/gray square, dashed line) as a function of $\eta$ in the residual state. Error bars represent the standard deviation in the residual state. Inset shows the angular average functions (a) $⟨f_n⟩\left(\theta \right)$ and $⟨f_{n^{\prime }}⟩\left(\theta \right)$ in black and green, respectively, and (b) $⟨f_t⟩\left(\theta \right)$ and $⟨f_{t^{\prime }}⟩\left(\theta \right)$ in red and blue, respectively, for $\eta =0.7$ calculated from the simulation data (points) together with the harmonic approximation (lines). Error bars represent the standard deviation in the residual state.

Figure 19

(Color online) Normalized shear stress ${\left(q/p\right)}^{\ast }$ in the residual state as a function of $\eta$ together with two analytical expressions given by Eq. (27). Error bars represent the standard deviation in the residual state.

Figure 20

(Color online) Map of radial forces for $\eta =0.2$ (up) and $\eta =0.7$ (down). Line thickness is proportional to the radial force. We represent the strong network in black and the weak network in red lines (gray) (see Sec. 6). The floating particles excluded from the force network are in white.

Figure 21

Proportion of sliding contacts as a function of $\eta$ averaged in the residual state. Error bars represent the standard deviation in the residual state.

Figure 22

(Color online) Probability distribution function of radial forces $f_{n^{\prime }}$ normalized by the average radial force $⟨f_{n^{\prime }}⟩$ in log-linear (up) and log-log (down) plots for different values of $\eta$.

Figure 23

(Color online) Probability distribution function of orthoradial forces $f_{t^{\prime }}$ normalized by the average orthoradial force $⟨|f_{t^{\prime }}|⟩$ in log-log for all values of $\eta$.

Figure 24

(Color online) Proportions of side-to-side $\left(ss\right)$, cap-to-side $\left(cs\right)$, and cap-to-cap $\left(cc\right)$ contacts as a function of $\eta$ in the residual state. Error bars represent the standard deviation in the residual state.

Figure 25

(Color online) Shear strength ${\left(q/p\right)}^{\ast }$ (circle) for $cs$ (square), $ss$ (triangle up), and $cc$ (triangle down) contacts as a function of $\eta$, together with the harmonic approximation fits in $\left(\mathit{n},\mathit{t}\right)$ frame (—) and $\left({\mathit{n}}^{\prime },{\mathit{t}}^{\prime }\right)$ frame $(\dots )$.

Figure 26

(Color online) Partial connectivity diagrams for all packings in the residual state.

Figure 27

(Color online) Partial contact orientation anisotropies $a_{ccc}$ (triangle up), $a_{ccs}$ (square), and $a_{css}$ (triangle down) of $cc$, $cs$, and $ss$ contacts as the function of $\eta$ in the residual state. Error bars represent the standard deviation in the residual state.

Figure 28

(Color online) Partial radial force anisotropies $a_{f{n}^{\prime }cc}$, $a_{f{n}^{\prime }cs}$, and $a_{f{n}^{\prime }ss}$ and partial orthoradial force anisotropies $a_{f{t}^{\prime }cc}$, $a_{f{t}^{\prime }cs}$, and $a_{f{t}^{\prime }ss}$ for different contact types as a function of $\eta$ in the residual state. Error bars represent the standard deviation in the residual state.

Figure 29

(Color online) Snapshot of radial forces for $\eta =0.2$ (up) and $\eta =0.7$ (down). Line thickness is proportional to the radial force. The cap-to-cap, cap-to-side, and side-to-side contacts are in black, in red (dark gray), and in green (light gray).

Figure 30

(Color online) Proportions of cap-to-cap $\left(k_{cc}\right)$, cap-to-side $\left(k_{cs}\right)$, and side-to-side $\left(k_{ss}\right)$ contacts in the strong (plain line) and weak (dashed line) networks as a function of $\eta$ in the residual state. Error bars represent the standard deviation in the residual state.