Comparison of compression versus shearing near jamming, for a simple model of athermal frictionless disks in suspension

Using a simplified model for a non-Brownian suspension, we numerically study the response of athermal, overdamped, frictionless disks in two dimensions to isotropic and uniaxial compression, as well as to pure and simple shearing, all at finite constant strain rates $\dot{\epsilon }$. We show that isotropic and uniaxial compression result in the same jamming packing fraction ${\varphi }_J$, while pure-shear- and simple-shear-induced jamming occurs at a slightly higher ${\varphi }_J^{*}$, consistent with that found previously for simple shearing. A critical scaling analysis of pure shearing gives critical exponents consistent with those previously found for both isotropic compression and simple shearing. Using orientational order parameters for contact bond directions, we compare the anisotropy of the force and contact networks at both lowest nematic order, as well as higher $2n$-fold order.

Figure 1

The geometry of the four linear deformations described by the strain rate tensors of Eqs. (5) and (6): (a) Uniaxial compression, (b) isotropic compression, (c) pure shear, and (d) simple shear.

Figure 2

(a) Pressure $p$ and (b) shear stress $\sigma$ vs packing $\varphi$ for uniaxial compression $(\square )$, isotropic compression $\left(▴\right)$, pure shear $(\circ )$, and simple shear $\left(⧫\right)$ of $N=32768$ particles, at different strain rates $\dot{\epsilon }={10}^{-4}$, ${10}^{-6}$, and ${10}^{-8}$. The vertical dashed line indicates the isotropic compression-driven jamming ${\varphi }_J=0.8415$. The estimated error in the data is typically smaller than the size of each data point symbol. For the shear stress in panel (b) ${\sigma }_{\mathrm{iso}}=0$ and is not shown.

Figure 3

(a) Average number of contacts per particle $Z$ vs $\varphi$ for uniaxial compression, isotropic compression, pure shear, and simple shear at the two strain rates $\dot{\epsilon }={10}^{-7}$ and ${10}^{-8}$. Rattler particles are included in the calculation of $Z$. (b) Fraction of the contacts that are between two small particles, two big particles, and a big and small particle, vs $\varphi$ for uniaxial compression, isotropic compression, pure shear, and simple shear at $\dot{\epsilon }={10}^{-7}$. The vertical dashed line indicates the isotropic compression-driven jamming ${\varphi }_J=0.8415$. The system has $N=32768$ particles. The estimated error in the data is typically smaller than the size of each data point symbol.

Figure 4

Stress ratios comparing uniaxial compression, isotropic compression, pure shear, and simple shear. Pressure ratios (a) $p_{\mathrm{uni}}/p_{\mathrm{iso}}$, (b) $p_{\mathrm{ss}}/p_{\mathrm{ps}}$, (c) $p_{\mathrm{uni}}/p_{\mathrm{ps}}$, and (d) shear stress ratio ${\sigma }_{\mathrm{uni}}/{\sigma }_{\mathrm{ps}}$ vs $\varphi$ for different strain rates $\dot{\epsilon }$. The vertical dashed line indicates the isotropic compression-driven jamming ${\varphi }_J=0.8415$. The system has $N=32768$ particles.

Figure 5

(a) Stress tensor anisotropy $\mu =\sigma /p$, and (b) fabric tensor anisotropy $\mathrm{\Delta }\lambda$ vs $\varphi$ for different strain rates $\dot{\epsilon }$, for uniaxial compression ($\circ$), pure shearing ($\square$), and simple shearing ($\times$). The strain rates vary from $\dot{\epsilon }={10}^{-4}$ to ${10}^{-8.5}$ as the curves go from top to bottom; for simple shear the slowest rate is ${10}^{-8}$. The vertical dashed line indicates the isotropic compression-driven jamming ${\varphi }_J=0.8415$. The system has $N=32768$ particles. The estimated error in the data is typically smaller than the size of each data point symbol.

Figure 6

Probability density $\mathcal{P}\left(\theta \right)$ for the system contact network to have a bond directed at angle $\theta$ with respect to the compressive direction $\widehat{\mathbf{x}}$. Results are plotted vs $\theta$ for several different packing fractions $\varphi$ at strain rate $\dot{\epsilon }={10}^{-7}$. (a) is for uniaxial compression, while (b) is for pure shearing. The solid horizontal black line at $\mathcal{P}=1/\pi$ represents the average value. The system has $N=32768$ particles.

Figure 7

Average force-moment $h=f_{ij}^{\mathrm{el}}{r}_{ij}$ per radian at contact bond angle $\theta$, $\tilde{h}\left(\theta \right)$, normalized by the average force moment $\langle h\rangle$, vs $\theta$ for several different packing fractions $\varphi$ at strain rate $\dot{\epsilon }={10}^{-7}$. (a) is for uniaxial compression, while (b) is for pure shearing. The solid horizontal black line at $\tilde{h}/\langle h\rangle =1/\pi$ represents the average value. The system has $N=32768$ particles.

Figure 8

Average force-moment on a contact bond at angle $\theta$, $h\left(\theta \right)=\tilde{h}\left(\theta \right)/\mathcal{P}\left(\theta \right)$, normalized by $\pi \langle h\rangle$, vs $\theta$ for several different packing fractions $\varphi$ at strain rate $\dot{\epsilon }={10}^{-7}$. Panel (a) is for uniaxial compression, while panel (b) is for pure shearing. The solid horizontal black line at $\tilde{h}/\pi \langle h\rangle =1/\pi$ represents the average value. The system has $N=32768$ particles.

Figure 9

(a) Orientational order parameter $S_{2n}$ for the contact network, and (b) orientational order parameter ${\mathbb{S}}_{2n}$ for the force network, vs $\varphi$ for strain rate $\dot{\epsilon }={10}^{-7}$. Results are shown for nematic ($n=1$), tetratic ($n=2$), hexatic ($n=3$), and eightfold ($n=4$) orientational order. $S_2=\mathrm{\Delta }\lambda$ is the same as the fabric tensor anisotropy, while ${\mathbb{S}}_2=\mu$ is the same as the stress tensor anisotropy. Closed symbols denote pure shearing while open symbols denote uniaxial compression. The vertical dashed line indicates the isotropic compression-driven jamming ${\varphi }_J=0.8415$. The system has $N=32768$ particles.

Figure 10

Anisotropic part of the contact network bond orientation probability $\mathrm{\Delta }\mathcal{P}\left(\theta \right)=\mathcal{P}\left(\theta \right)-1/\pi$, normalized by the magnitude of the nematic order Fourier coefficient $S_2$, vs $\theta$ for several different packing fractions $\varphi$ at strain rate $\dot{\epsilon }={10}^{-7}$. (a) is for uniaxial compression, while (b) is for pure shearing. The solid black line gives the functional form for the purely nematic term, $(2/\pi )cos2\theta$. The system has $N=32768$ particles.

Figure 11

Anisotropic part of the average force-moment orientation $\mathrm{\Delta }\tilde{h}\left(\theta \right)=\tilde{h}\left(\theta \right)-\langle h\rangle /\pi$, normalized by the magnitude of the nematic order Fourier coefficient $\langle h\rangle {\mathbb{S}}_2$, vs $\theta$ for several different packing fractions $\varphi$ at strain rate $\dot{\epsilon }={10}^{-7}$. Panel (a) is for uniaxial compression, while panel (b) is for pure shearing. The solid black line gives the functional form for the purely nematic term, $(2/\pi )cos2\theta$. The system has $N=32768$ particles.

Figure 12

(a) Ratio of orientational order parameters $S_{2n}/S_2$ for the bond directions in the geometrical contact network, and (b) ratio of orientational order parameters ${\mathbb{S}}_{2n}/{\mathbb{S}}_2$ for the force network, vs $\varphi$ for strain rate $\dot{\epsilon }={10}^{-7}$. Results are shown for tetratic ($n=2$), hexatic ($n=3$), and eightfold ($n=4$) orientational order. Closed symbols denote pure shearing while open symbols denote uniaxial compression. The vertical dashed line indicates the isotropic compression-driven jamming ${\varphi }_J=0.8415$. The system has $N=32768$ particles.

Figure 13

Probability density $\mathcal{P}\left(\theta \right)$ for the contact network to have a bond directed at angle $\theta$ with respect to the flow direction $\widehat{\mathbf{x}}$, in simple shearing. Results are plotted vs $\theta$ for several different packing fractions $\varphi$ at strain rate $\dot{\epsilon }={10}^{-7}$. The solid horizontal black line at $\mathcal{P}=1/\pi$ represents the average value. The system has $N=32768$ particles.

Figure 14

Average force-moment per radian $\tilde{h}\left(\theta \right)$, normalized by the average force moment $\langle h\rangle$, in simple shearing. Results are plotted vs bond angle $\theta$ for several different packing fractions $\varphi$ at strain rate $\dot{\epsilon }={10}^{-7}$. The solid horizontal black line at $\tilde{h}/\langle h\rangle =1/\pi$ represents the average value. The system has $N=32768$ particles.

Figure 15

Average force-moment $h\left(\theta \right)=\tilde{h}\left(\theta \right)/\mathcal{P}\left(\theta \right)$, normalized by $\pi \langle h\rangle$, in simple shearing. Results are plotted vs bond angle $\theta$ for several different packing fractions $\varphi$ at strain rate $\dot{\epsilon }={10}^{-7}$. The solid horizontal black line at $\tilde{h}/\pi \langle h\rangle =1/\pi$ represents the average value. The system has $N=32768$ particles.

Figure 16

(a) Orientational order parameter $S_{2n}$ for the contact network, and (b) orientational order parameter ${\mathbb{S}}_{2n}$ for the force network, for simple shearing (closed symbols) compared to pure shearing (open symbols). Results are plotted vs the packing $\varphi$ for strain rate $\dot{\epsilon }={10}^{-7}$, showing nematic ($n=1$), tetratic ($n=2$), hexatic ($n=3$), and eightfold ($n=4$) orientational order. The vertical dashed line indicates the shear-driven jamming ${\varphi }_J=0.8435$. The system has $N=32768$ particles.

Figure 17

Orientation angles ${\theta }_{2n}$ and ${\vartheta }_{2n}$ of the order parameters $S_{2n}$ and ${\mathbb{S}}_{2n}$ for simple shearing. Results are plotted vs the packing $\varphi$ for strain rate $\dot{\epsilon }={10}^{-7}$. The vertical dashed line represents the shear-driven jamming ${\varphi }_J^{*}=0.8435$. The horizontal dashed line is at $\theta =-\pi /4$. For the eightfold ordering ($n=4$) we do not show results for $\varphi >0.83$ since then $S_8$ and ${\mathbb{S}}_8$ are too small to determine ${\theta }_8$ and ${\vartheta }_8$ reliably. The system has $N=32768$ particles.

Figure 18

For simple shearing: Anisotropic part of the contact network bond orientation probability $\mathrm{\Delta }\mathcal{P}\left(\theta \right)=\mathcal{P}\left(\theta \right)-1/\pi$, normalized by the magnitude of the nematic order Fourier coefficient $S_2$, vs $\theta$ for several different packing fractions $\varphi$ at strain rate $\dot{\epsilon }={10}^{-7}$. The solid black line gives the functional form for the purely nematic term, $(2/\pi )cos2(\theta -{\theta }_2)$, where we take ${\theta }_2=-\pi /4$. The system has $N=32768$ particles.

Figure 19

For simple shearing: Anisotropic part of the average force-moment orientation $\mathrm{\Delta }\tilde{h}\left(\theta \right)=\tilde{h}\left(\theta \right)-\langle h\rangle /\pi$, normalized by the magnitude of the nematic order Fourier coefficient $\langle h\rangle {\mathbb{S}}_2$, vs $\theta$ for several different packing fractions $\varphi$ at strain rate $\dot{\epsilon }={10}^{-7}$. The solid black line gives the functional form for the purely nematic term, $(2/\pi )cos2(\theta -{\vartheta }_2)$, where we take ${\vartheta }_2=-\pi /4$. The system has $N=32768$ particles.

Figure 20

(a) Pressure $p$ vs packing $\varphi$ at a strain rate $\dot{\epsilon }={10}^{-7}$, for isotropic compression of $N=8192$ particles starting from different initial packings ${\varphi }_{\mathrm{init}}$. The vertical dashed line indicates the compression-driving jamming ${\varphi }_J=0.8415$. The width of each curve indicates the estimate error. (b) Data of panel (a) replotted as $p$ vs ${\varphi }_{\mathrm{init}}$, at several different packings $\varphi$. The solid lines are cubic polynomial fits. The black dots and dashed line in panel (a) are the extrapolated values of $p$ as ${\varphi }_{\mathrm{init}}\to 0$, obtained from such fits. Panels (c) and (d) are analogous plots of $p$ for the case of uniaxial compression, while panels (e) and (f) are the analogous plots of shear stress $\sigma$ for uniaxial compression.

Figure 21

(a) Average contact number $Z$ (including rattlers) vs packing $\varphi$ at a strain rate $\dot{\epsilon }={10}^{-7}$, for isotropic compression of $N=8192$ particles starting from different initial packings ${\varphi }_{\mathrm{init}}$. The vertical dashed line indicates the compression-driving jamming ${\varphi }_J=0.8415$. The width of each curve indicates the estimate error. (b) Data of panel (a) replotted as $Z$ vs ${\varphi }_{\mathrm{init}}$, at several different packings $\varphi$. The solid lines are cubic polynomial fits. The black dots and dashed line in panel (a) are the extrapolated values of $Z$ as ${\varphi }_{\mathrm{init}}\to 0$, obtained from such fits. Panels (c) and (d) are analogous plots of $Z$ for the case of uniaxial compression.

Figure 22

Pressure ratio $p_{\mathrm{uni}}/p_{\mathrm{iso}}$ vs packing $\varphi$, comparing uniaxial to isotropic compression, at a strain rate $\dot{\epsilon }={10}^{-7}$ for $N=8192$ particles starting from different initial packings ${\varphi }_{\mathrm{init}}$. (b) An expanded view of panel (a), looking closer in the vicinity of the jamming transition ${\varphi }_J$. The vertical dashed lines indicate the compression-driving jamming ${\varphi }_J=0.8415$. For clarity, data points are shown only at intervals of $\mathrm{\Delta }\varphi =0.01$ and representative error bars are shown only on a subset of those points.

Figure 23

(a) Pressure $p$, (b) shear stress $\sigma$, and (c) average contact number $Z$ (including rattlers) vs net strain $\epsilon =\dot{\epsilon }t$, as the system is pure sheared at $\varphi =0.80<{\varphi }_J$ with the rate $\dot{\epsilon }={10}^{-7}$. Panels (d), (e), and (f) show similar results at $\varphi =0.86>{\varphi }_J$. Results are averaged over 10 independent initial configurations. The thicker blue lines show results starting from initial configurations obtained from uniaxial compression, and so have a finite initial shear stress $\sigma >0$; the thinner red lines show results starting from initial configurations obtained from isotropic compression, and so have an initial $\sigma =0$. The system has $N=32768$ particles.

Figure 24

(a) Pressure $p$ and (b) shear stress $\sigma$ vs strain rate $\dot{\epsilon }$, in steady-state pure shearing, for several different values of packing $\varphi$ near jamming. The system has $N=32768$ particles.

Figure 25

(a) Pressure $p$ vs packing $\varphi$ in steady-state pure shear, for different strain rates $\dot{\epsilon }$. The vertical dashed line locates the jamming ${\varphi }_J^{*}=0.84319$. (b) Scaled pressure $p/{\dot{\epsilon }}^q$ vs scaled packing difference $(\varphi -{\varphi }_J^{*})/{\dot{\epsilon }}^{1/z\nu }$, for different strain rates $\dot{\epsilon }$, according to the scaling Eq. (B1). The scaling collapse is obtained using ${\varphi }_J^{*}=0.84319$, $q=0.305$, and $1/z\nu =0.268$. The system has $N=32768$ particles.

Figure 26

Critical parameters obtained by fitting the data of Fig. 25 to the scaling form of Eq. (B1), for different windows of data. Data windows are defined by $\dot{\epsilon }\le {\dot{\epsilon }}_{\max }$ and $\varphi \in [{\varphi }_{\min },{\varphi }_{\max }]$. (a) The critical jamming packing ${\varphi }_J^{*}$, (b) the ${\chi }^2$ per degree of freedom $n_f$ of the fit, (c) critical exponent $q$, (d) critical exponent $1/z\nu$, (e) bulk viscosity exponent $\beta =(1-q)z\nu$, and (f) yield pressure exponent $y=qz\nu$.