Emergent SO(3) Symmetry of the Frictionless Shear Jamming Transition

Abstract

We study the shear jamming of athermal frictionless soft spheres, and find that in the thermodynamic limit, a shear-jammed state exists with different elastic properties from the isotropically-jammed state. For example, shear-jammed states can have a non-zero residual shear stress in the thermodynamic limit that arises from long-range stress-stress correlations. As a result, the ratio of the shear and bulk moduli, which in isotropically-jammed systems vanishes as the jamming transition is approached from above, instead approaches a constant. Despite these striking differences, we argue that in a deeper sense, the shear jamming and isotropic jamming transitions actually have the same symmetry, and that the differences can be fully understood by rotating the six-dimensional basis of the elastic modulus tensor.

Appendix: Stress Correlation Functions

The scalings of p, $\sigma _{xy}$, $\sigma _{xz}$, B, $C_{xyxy}$ and $C_{xzxz}$ can be understood microscopically in terms of the behavior of spatial bond correlations. We can express the average deviatoric squared stresses $\tilde{\sigma }^2$, $p^2$, $\sigma _{xy}^2$ and $\sigma _{xz}^2$ in terms of associated stress correlation functions $C^{(\sigma )}(\mathbf {x}),C^{(p)}(\mathbf {x}),C^{(\sigma )}_{xy}(\mathbf {x})$ and $C^{(\sigma )}_{xz}(\mathbf {x})$.

In order to do so, we extend the correlation function $C^{(\sigma )}(\mathbf {x})$ defined in reference [24] to take into account the anisotropy caused by the shear. The stress tensor is defined through

$$\begin{aligned} \sigma _{\alpha \beta } = - \frac{1}{V} \sum _k^{N_b} b^{(k)}{\hat{r}^{(k)}_\alpha }{\hat{r}^{(k)}_\beta }\,, \end{aligned}$$

(2)

where the index k indicates a contact (bond) between two particles, $N_b$ is the number of bonds, $\mathbf {r}^{(k)}=\frac{\hat{r}^{(k)}}{|\mathbf {r}^{(k)}|}$ is the separation between the two touching particles, and $b^{(k)}=f^{(k)} |\mathbf {r}^{\,(k)}|$, where $f^{(k)}$ is the force of bond k. The product between generic components of the stress tensor is

$$\begin{aligned} \sigma _{\alpha \beta }\sigma _{\gamma \delta }&= \frac{1}{V^2}\sum _{k,k'} b^{(k)}b^{(k')}{\hat{r}^{(k)}_\alpha }{\hat{r}^{(k)}_\beta }\hat{r}^{(k')}_\gamma \hat{r}^{(k')}_\delta \nonumber \\&= \frac{1}{V^2}\sum _{k} {b^{(k)}}^2{\hat{r}^{(k)}_\alpha }{\hat{r}^{(k)}_\beta }\hat{r}^{(k)}_\gamma \hat{r}^{(k)}_\delta + \nonumber \\&\quad +\frac{1}{V^2}\sum _{k\ne k'}b^{(k)}b^{(k')}{\hat{r}^{(k)}_\alpha }{\hat{r}^{(k)}_\beta }\hat{r}^{(k')}_\gamma \hat{r}^{(k')}_\delta \end{aligned}$$

(3)

Taking out a factor $N_b$, both terms can be seen as an average over the bonds,

$$\begin{aligned} \sigma _{\alpha \beta }\sigma _{\gamma \delta }&= \frac{N_b}{V^2}\left\langle b^2\,{\hat{r}^{(k)}_\alpha }{\hat{r}^{(k)}_\beta }\hat{r}^{(k)}_\gamma \hat{r}^{(k)}_\delta \right\rangle +\nonumber \\&\quad + \frac{N_b}{V^2}\left\langle \sum _{k\ne 0} b^{(0)}b^{(k)}\,{\hat{r}^{(0)}_\alpha }{\hat{r}^{(0)}_\beta }\hat{r}^{(k)}_\gamma \hat{r}^{(k)}_\delta \right\rangle \,. \end{aligned}$$

(4)

From the second term in the right hand side (r.h.s.) we can define a correlation function

$$\begin{aligned} C^{(\sigma )}_{\alpha \beta \gamma \delta }(\mathbf {x})= \left\langle \sum _{k\ne 0} b^{(0)}b^{(k)}\left[ {\hat{r}^{(0)}_\alpha }{\hat{r}^{(0)}_\beta }{\hat{r}^{(k)}_\gamma }{\hat{r}^{(k)}_\delta }\right] \delta \left( \left[ \mathbf {x}^{(0)}-\mathbf {x}^{(k)}\right] -\mathbf {x}\right) \right\rangle \,, \end{aligned}$$

(5)

where $\mathbf {x}^{(k)}$ is the position of bond k.

The correlation function $C^{(\sigma )}_{\alpha \beta \gamma \delta }(\mathbf {x})$ is related to the stress through

$$\begin{aligned} \sigma _{\alpha \beta }\sigma _{\gamma \delta }= \frac{N_b}{V^2}\left\langle b^2\,{\hat{r}^{(k)}_\alpha }{\hat{r}^{(k)}_\beta }\hat{r}^{(k)}_\gamma \hat{r}^{(k)}_\delta \right\rangle + \frac{N_b}{V^2}\int d^dx \,C^{(\sigma )}_{\alpha \beta \gamma \delta }(\mathbf {x})\,. \end{aligned}$$

(6)

With an analogous procedure it is possible to define a wide variety of correlation functions,⁶ each with a relation that connects it to the stress.

In this instance we are interested in the correlation functions

$$\begin{aligned} C^{(\sigma )}(\mathbf {x})&= \left\langle \sum _{k\ne 0} b^{(0)}b^{(k)}\left[ (\hat{r}^{(0)}\cdot \hat{r}^{(k)})^2 -1/d\right] \delta \left( \left[ \mathbf {x}^{(0)}-\mathbf {x}^{(k)}\right] -\mathbf {x}\right) \right\rangle \,,\end{aligned}$$

(7)

$$\begin{aligned} C^{(p)}(\mathbf {x})&= \left\langle \sum _{k\ne 0} b^{(0)}b^{(k)}\left[ \frac{1}{d^2}\right] \delta \left( \left[ \mathbf {x}^{(0)}-\mathbf {x}^{(k)}\right] -\mathbf {x}\right) \right\rangle \,,\end{aligned}$$

(8)

$$\begin{aligned} C^{(\sigma )}_{xy}(\mathbf {x})&= \left\langle \sum _{k\ne 0} b^{(0)}b^{(k)}\left[ {\hat{r}^{(0)}_x}{\hat{r}^{(0)}_y}{\hat{r}^{(k)}_x}{\hat{r}^{(k)}_y}\right] \delta \left( \left[ \mathbf {x}^{(0)}-\mathbf {x}^{(k)}\right] -\mathbf {x}\right) \right\rangle \,,\end{aligned}$$

(9)

$$\begin{aligned} C^{(\sigma )}_{xz}(\mathbf {x})&= \left\langle \sum _{k\ne 0} b^{(0)}b^{(k)}\left[ {\hat{r}^{(0)}_x}{\hat{r}^{(0)}_z}{\hat{r}^{(k)}_x}{\hat{r}^{(k)}_z}\right] \delta \left( \left[ \mathbf {x}^{(0)}-\mathbf {x}^{(k)}\right] -\mathbf {x}\right) \right\rangle \,, \end{aligned}$$

(10)

that are related to the stress through

$$\begin{aligned} \tilde{\sigma }^2&= \frac{N_b}{V^2}\left\langle b^2\right\rangle \frac{d-1}{d} + \frac{N_b}{V^2}\int d^dx \,C^{(\sigma )}(\mathbf {x})\,,\end{aligned}$$

(11)

$$\begin{aligned} p^2&= \frac{N_b}{V^2d^2}\left\langle b^2\right\rangle + \frac{N_b}{V ^2}\int d^dx \,C^{(p)}(\mathbf {x})\,,\end{aligned}$$

(12)

$$\begin{aligned} \tilde{\sigma }_{xy}^2&= \frac{N_b}{V^2}\left\langle b^2\,\hat{r}_x^2\,\hat{r}_y^2 \right\rangle + \frac{N_b}{V ^2}\int d^dx \,C^{(\sigma )}_{xy}(\mathbf {x})\,,\end{aligned}$$

(13)

$$\begin{aligned} \tilde{\sigma }_{xz}^2&= \frac{N_b}{V^2}\left\langle b^2\,\hat{r}_x^2\,\hat{r}_z^2 \right\rangle + \frac{N_b}{V ^2}\int d^dx \,C^{(\sigma )}_{xz}(\mathbf {x})\,, \end{aligned}$$

(14)

The first term in the r.h.s. of Eqs. (6), (11), (12), (13), (14) is of order $\left<b^2\right>/N$, while the second term depends on the integral of the correlation function. Let $\mathcal {O}^2$ represent the l.h.s of any of these equations. If the corresponding correlation function is short-ranged then the integral is proportional to $\left<b^2\right> N^0$, and $\mathcal {O}^2\sim \left<b^2\right>/N \sim p^2/N$. However, if the correlation function is long-ranged, then the integral is proportional to $\left<b^2\right> N$, and

$$\begin{aligned} \mathcal {O}^2 \sim \left<b^2\right> N^0 \sim p^2 N^0 \end{aligned}$$

(15)

In Fig. 5a and b we show that the correlation function $C^{(\sigma )}(\mathbf {x})$ is short-ranged for isotropically-jammed states, and long-ranged for shear-jammed states: shear-jamming induces long-range correlations in the system, which lead to a non-zero deviatoric stress. The two bottom plots of Fig. 5 show that the correlation $C^{(\sigma )}_{xy}(\mathbf {x})$ along the shear is long-ranged, whereas in the orthogonal direction $C^{(\sigma )}_{xz}(\mathbf {x})$ is short-ranged, explaining why $\sigma _{xy}$ and $\sigma _{xz}$ scale differently.

Fig. 5

Bond correlations $C^{(\sigma )}(\mathbf {x}),C^{(\sigma )}_{xy}(\mathbf {x})$ and $C^{(\sigma )}_{xz}(\mathbf {x})$, normalized with the square pressure $p^2$, as a function of the distance x between bonds, for all system sizes at $\sigma _{xy}=10^{-8}$. The integrals of the correlation functions are related with the square stresses through Eqs. (11), (12), (13), (14): if the integral is extensive (i.e. the correlation in long-ranged) then the related squared stress is of order ${p^2}$. If it is intensive (short-range correlations), the squared stress is of order $p^2/N$. a Correlation $C^{(\sigma )}(\mathbf {x})$ in isotropically-jammed packings. It is short-ranged (inset), so $\tilde{\sigma }^2\sim {p^2}/N$ in such systems. b Correlation $C^{(\sigma )}(\mathbf {x})$ in shear-jammed packings. It does not decay to zero (inset), so $\tilde{\sigma }^2\sim p^2$ in such systems. c Correlation $C^{(\sigma )}_{xz}(\mathbf {x})$ in shear-jammed packings. There are no long-ranged correlations orthogonal to the direction of shear jamming (inset), so $\sigma _{xz}\sim {p^2}/N$. d Correlation $C^{(\sigma )}_{xy}(\mathbf {x})$ in shear-jammed packings. The long-distance correlations decay to a positive constant, so $\sigma _{xy}\sim p^2$. The insets depict a zoom of the same data of the larger plot, so that the differences between long-ranged correlation functions and short-ranged ones are more visible

The long-ranged nature of $C^{(\sigma )}_{xy}(\mathbf {x})$ leads to the scaling $\sigma _{xy}^2 \sim p^2$, consistent with Fig. 2. This in turn leads to the prediction that the scaling exponents for $C_{xyxy}$ and B are the same, consistent with Fig. 3.