Variable-amplitude oscillatory shear response of amorphous materials

Variable-amplitude oscillatory shear tests are emerging as powerful tools to investigate and quantify the nonlinear rheology of amorphous solids, complex fluids, and biological materials. Quite a few recent experimental and atomistic simulation studies demonstrated that at low shear amplitudes, an amorphous solid settles into an amplitude- and initial-conditions-dependent dissipative limit cycle, in which back-and-forth localized particle rearrangements periodically bring the system to the same state. At sufficiently large shear amplitudes, the amorphous system loses memory of the initial conditions, exhibits chaotic particle motions accompanied by diffusive behavior, and settles into a stochastic steady state. The two regimes are separated by a transition amplitude, possibly characterized by some critical-like features. Here we argue that these observations support some of the physical assumptions embodied in the nonequilibrium thermodynamic, internal-variables based, shear-transformation-zone model of amorphous viscoplasticity; most notably that “flow defects” in amorphous solids are characterized by internal states between which they can make transitions, and that structural evolution is driven by dissipation associated with plastic deformation. We present a rather extensive theoretical analysis of the thermodynamic shear-transformation-zone model for a variable-amplitude oscillatory shear protocol, highlighting its success in accounting for various experimental and simulational observations, as well as its limitations. Our results offer a continuum-level theoretical framework for interpreting the variable-amplitude oscillatory shear response of amorphous solids and may promote additional developments.

Figure 1

The amplitude of $m\left(t\right)$ in the oscillatory steady state, denoted by $m_0$, as a function of the strain amplitude ${\gamma }_0$, in the subyield regime.

Figure 2

$m\left(t\right)$ vs $\gamma \left(t\right)$ over one cycle in steady state for various ${\gamma }_0$'s (${\gamma }_0=0.0090,0.0095,0.0115,0.0125,0.0150,0.0200$ from the innermost curve to the outermost curve). For ${\gamma }_0<\tilde{\gamma }$, the amplitude of $m\left(t\right)$ is very small (recall that $\tilde{\gamma }\simeq 0.009$). For $\tilde{\gamma }<{\gamma }_0<{\gamma }_c$ the amplitude of $m\left(t\right)$ increases monotonically (cf. Fig. 1) until it saturates at unity (recall that ${\gamma }_c\simeq 0.01155$). For ${\gamma }_0>{\gamma }_c$, finite time yielding periods, in which $\left|m\right|=1$ changes to $\left|m\right|<1$, are observed. The duration of these periods increases with increasing ${\gamma }_0$.

Figure 3

$\chi$ vs the accumulated strain ${\gamma }_{acc}\equiv {\int }_0^t\left|\dot{\gamma }\right|dt$ for various ${\gamma }_0\gtrsim {\gamma }_c$, for both ${\chi }_0=0.0569={\chi }_{\infty }-0.0207<{\chi }_{\infty }$ and ${\chi }_0=0.0983={\chi }_{\infty }+0.0207>{\chi }_{\infty }$ (the lowest curve corresponds to ${\gamma }_0=0.0115$ and the next to it to ${\gamma }_0=0.0125$. The rest of the curves follow the legend). For sufficiently large ${\gamma }_0$'s, the fixed point ${\chi }_{\infty }=0.0776$ is reached within the range of ${\gamma }_{acc}$ shown. For smaller ${\gamma }_0$'s, closer to ${\gamma }_c$, the fixed point is not yet reached.

Figure 4

(a) The relaxation time to steady state ${\tau }_{\chi }$ as a function of ${\gamma }_0$. The dashed vertical line corresponds to ${\gamma }_0={\gamma }_c$. (b) The same data as in (a), but on a log-log scale. The solid line has a slope of $-1.1$, which is consistent with the theoretical prediction in Eq. (19), ${\tau }_{\chi }\sim {\left({\gamma }_0-{\gamma }_c\right)}^{-1}$.

Figure 5

The normalized time ${\tau }_m(m=0.99,s_0)$ [where the thermal activation rate factor is suppressed, cf. Eq. (18)] it takes an initially isotropic system, $m(t=0)$, to reach $m=0.99$ as a function of $s_0$, the magnitude of an applied unidirectional (i.e., not oscillatory) shear stress in the subyield regime, $s_0<1$.

Figure 6

Main: The logarithm of the dissipation per cycle in steady state ${log}_{10}\left(\mathcal{A}\right)$ vs ${\gamma }_0$, both in the subyield regime close to the transition (${\gamma }_0\lesssim {\gamma }_c$) and in the postyield regime (${\gamma }_0>{\gamma }_c$). Inset: The same as the main panel, but with $\mathcal{A}$ being plotted in linear scale. In the inset it is observed that while the transition is sharp, it is still continuous and differentiable. Above the transition, but not too close to it, $\mathcal{A}$ approximately follows $\mathcal{A}\sim {\gamma }_0-{\gamma }_c$.

Figure 7

Stress-strain curves for various ${\gamma }_0$'s, ranging from below ${\gamma }_c$ (${\gamma }_0=0.0095$, innermost curve) to well above it (${\gamma }_0=0.02$, outermost curve).