Softening and Residual Loss Modulus of Jammed Grains under Oscillatory Shear in an Absorbing State

From a theoretical study of the mechanical response of jammed materials comprising frictionless and overdamped particles under oscillatory shear, we find that the material becomes soft, and the loss modulus remains nonzero even in an absorbing state where any irreversible plastic deformation does not exist. The trajectories of the particles in this region exhibit hysteresis loops. We succeed in clarifying the origin of the softening of the material and the residual loss modulus with the aid of Fourier analysis. We also clarify the roles of the yielding point in the softening to distinguish the plastic deformation from reversible deformation in the absorbing state.

Figure 1

Nonaffine particle trajectories in the last two cycles for ${\gamma }_0=0.02$ (a) and 0.1 (b) with $\omega ={10}^{-4}{\tau }_0^{-1}$ and $\varphi =0.87$, which corresponds to $\varphi -{\varphi }_J=0.029$. The circles represent the trajectory in the last cycle. The line represents the trajectory in the second to the last cycle.

Figure 2

Storage modulus $G^{\prime }$ obtained in our simulation (filled symbols) against ${\gamma }_0$ for $\omega ={10}^{-4}{\tau }_0^{-1}$ with $\varphi =0.870$, 0.860, 0.850, and 0.845, which corresponds to $\varphi -{\varphi }_J=0.029$, 0.019, 0.009, and 0.004, respectively. The legends represent the packing fraction $\varphi$. The data in the absorbing (plastic) state obtained in our simulation are shown in larger (smaller) filled symbols. The open pentagons represent the yielding strain amplitude ${\gamma }_c$, while other open symbols represent the theoretical expression using $G_T^{\prime }$ in Eq. (14). (Inset) Scaled storage modulus ${\tilde{G}}^{\prime }=G^{\prime }/\sqrt{\varphi -{\varphi }_J}$ obtained in our simulation (filled symbols) and its theoretical expression using $G_T^{\prime }$ (open symbols) in Eq. (14) against scaled strain amplitude ${\tilde{\gamma }}_0={\gamma }_0/(\varphi -{\varphi }_J)$ in the absorbing state.

Figure 3

(a) Loss modulus $G^{\prime \prime }$ in the absorbing state obtained in our simulation (filled symbols) and its theoretical expression $G_T^{\prime \prime }$ (open symbols) in Eq. (15) against ${\gamma }_0$ for $\omega ={10}^{-4}{\tau }_0^{-1}$ with $\varphi =0.870$, 0.860, 0.850, and 0.845, which corresponds to $\varphi -{\varphi }_J=0.029$, 0.019, 0.009, and 0.004, respectively. (b) Loss modulus $G^{\prime \prime }$ against $\omega {\tau }_0$ for $\varphi =0.87$ with ${\gamma }_0=0.01$.

Figure 4

Schematics of the nonaffine trajectory when only ${\mathit{a}}_i^{(1)}$ is nonzero (a) and only ${\mathit{a}}_i^{(1)}$ and ${\mathit{b}}_i^{(1)}$ are nonzero (b).

Figure 5

(a) Magnitudes of Fourier coefficients $a^{(n)}$ and $b^{(n)}$ against $n$ for $\varphi =0.87$ and ${\gamma }_0=0.02$ with $\omega {\tau }_0={10}^{-4}$ (filled symbols) and ${10}^{-5}$ (open symbols). (b) Magnitudes of the Fourier coefficients $a^{(n)}$ and $b^{(n)}$ normalized by ${\gamma }_0$ against ${\gamma }_0$ for $\varphi =0.87$ and $\omega {\tau }_0={10}^{-4}$ with $n=1$. $\varphi =0.87$ corresponds to $\varphi -{\varphi }_J=0.029$.