Eigenvalue analysis of stress-strain curve of two-dimensional amorphous solids of dispersed frictional grains with finite shear strain

The stress-strain curve of two-dimensional frictional dispersed grains interacting with a harmonic potential without considering the dynamical slip under a finite strain is determined by using eigenvalue analysis of the Hessian matrix. After the configuration of grains is obtained, the stress-strain curve based on the eigenvalue analysis is in almost perfect agreement with that obtained by the simulation, even if there are plastic deformations caused by stress avalanches. Unlike the naive expectation, the eigenvalues in our model do not indicate any precursors to the stress-drop events.

Figure 1

A stress-strain curve for $0\le \gamma \le 0.5$ for one sample of the collection of grains ($N=128$), which includes the theoretical results (line) and simulation results (filled symbols) under the $\text{condition}\mathrm{\Delta }{\gamma }_{\text{Th}}=\mathrm{\Delta }{\gamma }_{\text{in}}={10}^{-4}$. The inset is a close-up of the stress-strain curve in the vicinity of a stress-drop event.

Figure 2

Stress-strain relations for $\mathrm{\Delta }{\gamma }_{\text{Th}}=1.0\times {10}^{-4}$ (blue circles) and $\mathrm{\Delta }{\gamma }_{\text{Th}}=1.0\times {10}^{-8}$ (red triangles) with $N=128$.

Figure 3

Plots of (a) the smallest eigenvalue except for the zero modes and (b) the rigidity $g$ based on the eigenvalue analysis (open symbols) and numerical shear stress (filled symbols) against $\gamma$ for $0.3408\le \gamma \le 0.3413$ and $N=128$.

Figure 4

Plots of eigenvectors at (a) ${\gamma }_{c-}$ and at (b) ${\gamma }_{c+}$ corresponding to the smallest eigenvalue. For visualization, the magnitudes of the vectors are three times larger than their true values ($N=1024$).

Figure 5

Plots of $|d\mathring{q}/d\gamma \rangle$ at $\gamma ={\gamma }_{c+}$ for $N=1024$ and $\mathrm{\Delta }{\gamma }_{\text{Th}}=1.0\times {10}^{-8}$, where (a) and (b) are based on the eigenvalue analysis and simulation, respectively. For visualization, we magnify the magnitudes of the vectors with the factor 10.0.

Figure 6

Plot of $\mathrm{\Delta }\mathring{q}{|}_c$ in the simulation for $N=1024$ and $\mathrm{\Delta }{\gamma }_{\text{Th}}=1.0\times {10}^{-8}$. For visualization, we magnify $\mathrm{\Delta }\mathring{q}{|}_c$ with the factor $1.0\times {10}^3$.

Figure 7

Plots of the shear modulus with (a) the theoretical evaluation (open symbols) and the simulation results (filled symbols) except for critical strain with the close-up of $g$ near a yielding point (inset), and (b) the smallest eigenvalue, except for zero modes against $\gamma$ for $0\le \gamma \le 0.002$ for $N=128$. Note that the rigidity is not plotted at the yielding points, because it diverges there.

Figure 8

The plot of the stress-strain curve in the region $0\le \gamma \le 0.002$ for $N=128$.

Figure 9

Plots of the rigidity $G$ based on the eigenvalue analysis (line) and on the simulation (filled symbols) with $\mathrm{\Delta }{\gamma }_{\text{Th}}=1.0\times {10}^{-4}$. We have used 30 samples for $N=128$, where the error bars represent the standard deviations for $\gamma$. Note that we have omitted the data if stress-drop events occur.

Figure 10

Plot of the maximum $\text{Err}$ against $\gamma$.

Figure 11

The stress-strain curve for $N=128$, which are the stress drop points for various $\mathrm{\Delta }{\gamma }_{\mathrm{Th}}$ (from $\mathrm{\Delta }{\gamma }_{\mathrm{Th}}={10}^{-4}$ and $\mathrm{\Delta }{\gamma }_{\mathrm{Th}}={10}^{-10}$).

Figure 12

Plot of the critical strain ${\gamma }_c$ against $\mathrm{\Delta }{\gamma }_{\mathrm{Th}}$ for $N=128$.

Figure 13

The plot of the smallest nonzero eigenvalue in the vicinity of ${\gamma }_c$ for $\mathrm{\Delta }{\gamma }_{\mathrm{Th}}=1.0\times {10}^{-10}$.

Figure 14

A schematic of the case where the particle $i$ interacts with the particle $j$ through the mirror image in $x$ direction.

Figure 15

A schematic of the case that the particle $i$ interacts with the particle $j$ through the mirror image in $y$ direction.

Figure 16

The plot of the rigidity $G_0$ in the linear response regime for various $k_T/k_N$ and $\varphi$, where open symbols and filled symbols are results of the theory and simulation, respectively.