Elasticity of frictionless particles near jamming

We study the linear elastic response of harmonic disk packings near jamming via three types of probes: (i) point forcing, (ii) constrained homogeneous deformation of subregions of large systems, and (iii) unconstrained deformation of the full system subject to periodic boundary conditions. For the point forcing, our results indicate that the transverse component of the response is governed by a lengthscale ${\xi }_T$, which scales with the confining pressure, $p$, as ${\xi }_T\sim p^{-0.25}$, while the longitudinal component is governed by ${\xi }_L$, which scales as ${\xi }_L\sim p^{-0.4}$. The former scaling is precisely the transverse lengthscale, which has been invoked to explain the structure of normal modes near the density of states anomaly in sphere packings, while the latter is much closer to the rigidity length, $l_{*}\sim p^{-0.5}$, which has been invoked to describe the jamming scenario. For the case of constrained homogeneous deformation, we find that $\mu \left(R\right)$, the value of the shear modulus measured in boxes of size $R$, gives a value much higher than the continuum result for small boxes and recedes to its continuum limit only for boxes bigger than a characteristic length, which scales like $p^{-0.5}$, precisely the same way as $l_{*}$. Finally, for the case of unconstrained homogeneous deformation, we find displacement fields with power spectra, which are consistent with independent, uncorrelated Eshelby transformations. The transverse sector is amazingly invariant with respect to $p$ and very similar to what is seen in Lennard-Jones glasses. The longitudinal piece, however, is sensitive to $p$. It develops a plateau at long wavelength, the start of which occurs at a length that grows in the $p\to 0$ limit. Strikingly, the same behavior is observed both for applied shear and dilation.

Figure 1

Arrows represent the displacements in response to a point force.

Figure 2

Maps of $q^4{s}_L\left(\mathbf{\text{q}}\right)$ and $q^4{s}_T\left(\mathbf{\text{q}}\right)$ using a decimal log scale plotted for $\varphi =0.85$ and $\varphi =0.925$.

Figure 3

$q^4{s}_L\left(q\right)$ for cuts along the axis at $\theta =0$ and $\theta =\pi /2$ at $\varphi =0.85$ (left) and $\varphi =0.925$ (right) compared with the continuum solution (dashed lines). Note that the continuum solution predicts precisely zero power for $\theta =0$.

Figure 4

$q^4{s}_T\left(q\right)$ for cuts along the axis at $\theta =0$ and $\theta =\pi /2$ at $\varphi =0.85$ (left) and $\varphi =0.925$ (right) compared with the continuum solution (dashed lines). Note that the continuum solution predicts precisely zero power for $\theta =\pi /2$.

Figure 5

$S_L\left(q\right)$ versus $q/2\pi$ (left) and $S_T\left(q\right)$ versus $q/2\pi$ (right) for different $\varphi$.

Figure 6

$S_L\left(q\right)$ versus $p^{-0.4}q/2\pi$ (left) and $S_T\left(q\right)$ versus $p^{-0.25}q/2\pi$ (right) for different $\varphi$.

Figure 7

Shear modulus $\mu \left(R\right)$ plotted against $R$ at different $\varphi$ (left) and ${\mu }_{\text{Born}}$ and $\mu$ vs. $p$ (right). The dashed lines indicate ${\mu }_{\text{Born}}(R\to 0)$ and $\mu (R\to \infty )$.

Figure 8

$\mu \left(R\right)/\mu -1$ plotted against $p^{0.5}R$ at different $\varphi$. In the inset, $\mu \left(R\right)/\mu -1$ vs. $R$ is shown. The dashed line shows $R^{-1}$.

Figure 9

(a) The applied forces $F_{i\alpha }^{\text{ext}}$ and (b) the corresponding response $u_{i\alpha }$ for a typical small system.

Figure 10

Dependence of the shear modulus $\mu \left({\lambda }^{\prime }\right)$ (left) and the relative error on the shear modulus $1-\mu \left({\lambda }^{\prime }\right)/\mu$ (right) on the wavelength ${\lambda }^{\prime }=2\pi /q^{\prime }$ for various $\varphi$. The inset is the same as the main plot but rescaled by ${\lambda }^{\prime 2}$. The dashed lines represent $\mu$.

Figure 11

Real-space images of dilatancy, $\mathrm{\Phi }$ (top, a and b), and vorticity, $\omega$ (bottom, c and d), generated for imposed compression (left, a and c) and shear (right, b and d) at $\varphi =0.925$.

Figure 12

Real-space images of dilatancy, $\mathrm{\Phi }$ (top, a and b), and vorticity, $\omega$ (bottom, c and d), generated for imposed compression (left, a and c) and shear (right, b and d) at $\varphi =0.85$.

Figure 13

Maps of the longitudinal (top) and transverse (bottom) scaled power spectrum, $q^{2}s$ in dilation (left) and shear (right) plotted for $\varphi =0.925$ using a decimal log scale.

Figure 14

Maps of the longitudinal (top) and transverse (bottom) scaled power spectrum, $q^{2}s$ in dilation (left) and shear (right) plotted for $\varphi =0.85$ using a decimal log scale.

Figure 15

$q^2{s}_L^s\left(q\right)$ (left) and $q^2{s}_T^s\left(q\right)$ (right) for cuts along the axis at $\theta =0$ and $\theta =\pi /4$ at $\varphi =0.925$. The insets plot $s_L^s\left(q\right)$ and $s_T^s\left(q\right)$.

Figure 16

$q^2{s}_L^s\left(q\right)$ (left) and $q^2{s}_T^s\left(q\right)$ (right) for cuts along the axis at $\theta =0$ and $\theta =\pi /4$ at $\varphi =0.85$. The insets plot $s_L^s\left(q\right)$ and $s_T^s\left(q\right)$.

Figure 17

$q^2{s}_L^c\left(q\right)$ vs. $q/2\pi$ (a) and $q^2{s}_T^c\left(q\right)$ vs. $q/2\pi$ (b) for different $\varphi$ in dilation. In the insets, $s_L^c\left(q\right)$ and $s_T^c\left(q\right)$ are plotted against $q/2\pi$.

Figure 18

$q^2{s}_L^s\left(q\right)$ vs. $q/2\pi$ (a) and $q^2{s}_T^s\left(q\right)$ vs. $q/2\pi$ (b) for different $\varphi$ in shear. In the insets, $s_L^s\left(q\right)$ and $s_T^s\left(q\right)$ are plotted against $q/2\pi$.