Slow Fatigue and Highly Delayed Yielding via Shear Banding in Oscillatory Shear

We study theoretically the dynamical process of yielding in cyclically sheared amorphous materials, within a thermal elastoplastic model and the soft glassy rheology model. Within both models we find an initially slow accumulation, over many cycles after the inception of shear, of low levels of damage in the form strain heterogeneity across the sample. This slow fatigue then suddenly gives way to catastrophic yielding and material failure. Strong strain localization in the form of shear banding is key to the failure mechanism. We characterize in detail the dependence of the number of cycles $N^{*}$ before failure on the amplitude of imposed strain, the working temperature, and the degree to which the sample is annealed prior to shear. We discuss our finding with reference to existing experiments and particle simulations, and suggest new ones to test our predictions.

Figure 1

SGR model. (a) Root mean square stress and (b) mean degree of shear banding over each cycle versus cycle number $N$ for strain amplitudes ${\gamma }_0=1.00,1.25,\dots ,2.75$ in curve sets with drops in (a) and rises in (b) right to left. Each curve within a set corresponds to a different random initial condition. $t_w={10}^7$, $T=0.3$. (c) Cycle number at failure $N^{*}$, (d) magnitude of stress drop $\mathrm{\Delta }\mathrm{\Sigma }$, and (e) final degree of shear banding ${⟨\dot{\gamma }⟩}_{\mathrm{f}}$ vs strain amplitude ${\gamma }_0$ for waiting times $t_w={10}^2,{10}^3,\dots ,{10}^7$ in curves bottom to top. Panel (c) only shows samples with $\mathrm{\Delta }\mathrm{\Sigma }>0.1$. $N^{*}$, $\mathrm{\Delta }\mathrm{\Sigma }$, ${⟨\dot{\gamma }⟩}_{\mathrm{f}}$ averaged over initial condition.

Figure 2

SGR model. (a) Root mean square stress and (b) mean degree of shear banding over each cycle as a function of cycle number $N$ for waiting time $t_w={10}^1,{10}^2,\dots ,{10}^7$ in curve sets with drops in (a) and rises in (b) left to right. ${\gamma }_0=1.5$, $T=0.3$. (c) Cycle number at failure $N^{*}$, (d) magnitude of stress drop $\mathrm{\Delta }\mathrm{\Sigma }$, and (e) final degree of shear banding ${⟨\dot{\gamma }⟩}_{\mathrm{f}}$ vs waiting time, $t_{\mathrm{w}}$. Strain amplitude ${\gamma }_0=1.125,1.250,1.375,\dots ,2.250$ in curves blue to orange, i.e., top to bottom in (c), bottom to top at right of (d), and with ${\gamma }_0=1.125,1.25\dots 1.375$ bottom up and 1.5,...2.25 top down in (e). Panel (c) only shows cases for which $\mathrm{\Delta }\mathrm{\Sigma }>0.1$.

Figure 3

SGR model. (a) cycle number at failure $N^{*}$ and (b) stress drop $\mathrm{\Delta }\mathrm{\Sigma }$ as a function of waiting time $t_{\mathrm{w}}$ and strain amplitude ${\gamma }_0$. In the white region, no yielding occurs.

Figure 4

TEP model. (a) Root mean square stress and (b) mean degree of shear banding over each cycle as a function of cycle number $N$ for strain amplitudes ${\gamma }_0=0.90,0.95,\dots ,1.50$ in curve sets with drops in (a) and rises in (b) right to left. $T_0=0.01$, $T=0.007$. Cycle number at yielding $N^{*}$ vs strain amplitude ${\gamma }_0$ for (c) prequench temperatures $T_0=0.001,0.002,\dots ,0.010$ in curves right to left at working temperature $T=0.001$ and (d) working temperatures $T=0.001,0.002,\dots ,0.010$ in curves right to left at prequench temperature $T_0=0.01$. Solid lines in (c)+(d) are fits to $N^{*}=A/({\gamma }_0-{\gamma }_c)$. Insets show ${\gamma }_c$ (symbols) fit (lines) to (c) ${\gamma }_c=B-C\sqrt{T_0}$ and (d) ${\gamma }_c=DT-E$.

Figure 5

TEP model. (a) Root mean square stress and (b) mean degree of shear banding over each cycle as a function of cycle number $N$ for prequench temperatures $T_0=0.001,0.002,\dots ,0.010$ in curves with drops in (a),(c) and rises in (b),(d) right to left. ${\gamma }_0=1.15$, $T=0.001$. (c)+(d) Counterpart curves for working temperatures $T=0.001,0.002,\dots ,0.010$ in curves turquoise to magenta. ${\gamma }_0=1.05$, $T_0=0.01$. (e) Cycle number at yielding $N^{*}$ vs prequench temperature $T_0$ for strain amplitudes ${\gamma }_0=1.10$, 1.15, 1.17, 1.20, 1.22 in curves downward. $T=0.001$. Solid lines: fits to $N^{*}=A{e}^{B/T_0}$. (f) $N^{*}$ vs working temperature $T$ for ${\gamma }_0=1.00$, 1.05, 1.07, 1.10, 1.15 in curves downward. $T_0=0.01$. Solid lines: fits to $N^{*}=C{e}^{D/T}$.