Asymmetric velocity correlations in shearing media

A model of soft frictionless disks in two dimensions at zero temperature is simulated with a shearing dynamics to study various kinds of asymmetries in sheared systems. We examine both single particle properties, the spatial velocity correlation function, and a correlation function designed to separate clockwise and counterclockwise rotational fields from one another. Among the rich and interesting behaviors we find that the velocity correlation along the two different diagonals corresponding to compression and dilation, respectively, are almost identical and, furthermore, that a feature in one of the correlation functions is directly related to irreversible plastic events.

Figure 1

Configuration with 4096 particles color coded according to the elastic energy of each particle. The dark particles (high elastic energy) make up force chains that preferably extend along the $\widehat{x}-\widehat{y}$ direction.

Figure 2

Measure of the anisotropy in both particle velocities and contact forces from simulations with $\dot{\gamma }={10}^{-7}$. Panel (a) shows the normalized $⟨v_x{v}_y⟩$ for individual particles versus density. The correlation changes from negative at low densities to positive around $\varphi \approx {\varphi }_J$ (dashed vertical line), signifying a change from a predominance of motion along the force chains to a slight overweight for motion perpendicular to the force chains. Panel (b) shows that the corresponding contact force correlation is always negative, which is consistence with the existence of force chains at all densities. This contact force correlation also has a marked feature at $\varphi \approx {\varphi }_J$.

Figure 3

(Color online) The anisotropy measured through the $⟨v_y^2⟩$ relative to $⟨{\mathbf{v}}^2⟩=⟨v_x^2⟩+⟨v_y^2⟩$. This fraction is close to 50% in the regions around jamming but decreases slightly with increasing density.

Figure 4

(Color online) Total velocity correlation along the four different main directions. The decay of the correlations is monotonic along the main directions, $\widehat{x}$ and $\widehat{y}$ and nonmonotonic along the diagonals. This is for $\varphi =0.8433\approx {\varphi }_J$ and shear rate $\dot{\gamma }={10}^{-6}$.

Figure 5

(Color online) Transverse and longitudinal velocity correlations, respectively, along the four different directions. The transverse correlation in panel (a) is very similar for all four directions whereas the longitudinal correlation function shown in panel (b) is different for the diagonal directions, compared to the two main directions. The differences in the total velocity correlations in Fig. 4 are thus related to differences in the longitudinal correlations.

Figure 6

(Color online) The mixed correlation functions $g_{yx}\left(r\widehat{s}\right)$ and $-g_{yx}\left(r\widehat{t}\right)$. The difference between the mixed correlation functions (apart from the trivially different sign) is yet an example of the asymmetry in the sheared system. The arrows in the inset illustrate the rotational velocity fields which have the effect that a velocity $v_y\left(0\right)>0$ at the origin on the average gives $v_x\left(r\widehat{s}\right)>0$ and $v_x\left(r\widehat{t}\right)<0$.

Figure 7

(Color online) Development of the asymmetry in $G_{\perp }^{+}\left(x\right)$. The function is nearly symmetric at $\varphi =0.82$ but develops a very pronounced asymmetry as the density is increased above ${\varphi }_J$. The solid lines are the symmetric $G_{\perp }$. The shear rate is $\dot{\gamma }={10}^{-7}$.

Figure 8

(Color online) Location of the respective minima of the correlation functions. ${\ell }_{+}$ and ${\ell }_{-}$ are the positions of the minima of $G_{\perp }^{+}$ at $x>0$ and $x<0$, respectively, whereas ${\ell }_{\text{sym}}$ is from the minimum of $G_{\perp }$. The asymmetry of $G_{\perp }^{+}$ which is reflected in the difference ${\ell }_{+}-{\ell }_{-}$ develops around ${\varphi }_J$.

Figure 9

(Color online) The correlation function $G_{\perp }^{+}\left(x\right)$ at $\varphi =0.8433\approx {\varphi }_J$ and several different shear rates down to $\dot{\gamma }={10}^{-8}$. The asymmetry increases slowly with decreasing shear rate.

Figure 10

(Color online) Position of the minima of $g_{\perp }^{+}\left(x\right)$ and $g_{\perp }$, respectively, at $\varphi =0.8433\approx {\varphi }_J$ and four different shear rates. Note the algebraic increase of ${\ell }_{+}\sim {\dot{\gamma }}^{-0.25}$ which would imply a divergence at vanishing $\dot{\gamma }$.

Figure 11

(Color online) Splitting of the correlation function into two different contributions at $\varphi =0.8433$ and $\dot{\gamma }={10}^{-7}$. The total $G_{\perp }^{+}\left(x\right)$ (open circles) is split into contributions from slow and fast particles, respectively. It is clear that the asymmetry is largely an effect of the fast particles which we relate to the plastic processes. The inset shows an idealized velocity profile for a typical plastic event with quadrupolar structure. Note that this velocity field is a consequence of our shearing geometry which is equivalent to a compression along the $\pm \left(x-y\right)$ directions together with an dilation in the orthogonal $\pm \left(x+y\right)$ directions.