Inverse Bauschinger to Bauschinger Crossover under Steady Shear in Amorphous Solids

TL;DR

This work investigates how directional memory in amorphous solids under steady shear can invert from the conventional Bauschinger effect (BE) to the inverse Bauschinger effect (IBE). Using 2D polydisperse glasses with swap-MC annealing across a range of $T_p$ and finite strain rates $\\dot{\\gamma}$, the authors identify a critical deformation history $\\gamma_{N,\\mathrm{crit}}(T_p,\\dot{\\gamma})$ that marks the BE-IBE transition, and construct a phase diagram in $(T_p,\\dot{\\gamma},\\gamma_N)$. Microscopic analysis via $D^2_{\\min}$ reveals that IBE correlates with network-like shear bands and rapid plastic healing upon reversal, while BE aligns with persistent localization and cumulative damage. The results position directional memory as an order parameter for a brittle-ductile crossover and illuminate how preparation and driving set the yielding mode, with implications for experiments and materials design.

Abstract

Directional memory in amorphous solids is commonly quantified through the Bauschinger effect, yet the observation of the inverse Bauschinger effect suggests that the sign of memory can invert, pointing to distinct underlying plastic organization. Here, we connect directional memory to the nature of yielding in steadily sheared amorphous solids. Using simulations of two-dimensional polydisperse glasses, we show that the type of directional memory (Bauschinger versus inverse Bauschinger) is jointly controlled by deformation history, strain rate, and parent temperature. We identify a critical history amplitude $γ_{N,\mathrm{crit}}(T_p,\dotγ)$ and construct a phase diagram that delineates regimes with memory inversion from those showing only conventional Bauschinger response. Microscopically, memory inversion correlates with network-like shear-band morphology and plastic healing, whereas conventional memory is associated with persistent localization and cumulative damage. These results establish directional memory as an order parameter for a shear-rate and annealing-controlled brittle-ductile crossover and suggest that plastic healing provides a generic route to memory inversion in disordered solids.

Figures (7)

• Figure 1: This figure depicts the shear-reversal protocol, where a sample prepared at $T_p=0.035$ with $N=64{,}000$ is first sheared in the forward direction to different $\gamma_N$ values (denoted by circles) at a strain rate of $5\times10^{-3}$. The sample is then sheared in the reverse direction until it reaches a zero-stress configuration, labeled $\sigma_0$, and upon further shear, it reaches zero strain at $\gamma_0$, where the stress $\sigma(\gamma_0)$ is, in general, non-zero once the system has yielded.
• Figure 2: Transition from inverse (IBE) to Bauschinger effect (BE). Panel (a) shows a symmetric stress-strain response for a freshly prepared, well-annealed glass sample ($T_p=0.035$) sheared at a strain rate $\dot{\gamma}=5\times10^{-3}$. Panels (b-f) show the evolution of directional memory with increasing strain history. Panels (b) and (c) show pronounced yielding in the reverse direction at $\gamma_N=0.12$ and $\gamma_N=0.15$, respectively. As $\gamma_N$ approaches $0.24$ in (d), the difference between reverse and forward responses decreases. With a further increase in strain history to $\gamma_N=0.48$, the reverse yield stress has fallen below the forward yield stress, and panel (f) at $\gamma_N=0.96$ shows this more prominently, corresponding to the classical Bauschinger effect.
• Figure 3: Bauschinger effect. Panel (a) shows a symmetric stress-strain response for a freshly prepared glass sample ($T_p=0.20$) sheared at a strain rate $\dot{\gamma}=5\times10^{-3}$. Panels (b-c) show the evolution of directional memory with increasing strain history for $\gamma_N=0.096$, $\gamma_N=0.24$ (inset of (b)), and $\gamma_N=0.48$. In all cases, the response is asymmetric, with the reverse yield stress always smaller than the forward yield stress (i.e., the direction of the previous shear), corresponding to the classical Bauschinger effect.
• Figure 4: Annealing dependence of directional memory. (a) The panel shows the difference between the yield stresses measured when the system is sheared in the previously sheared direction to a maximum strain $\gamma_N$, $\sigma_Y(p)$, and in the opposite direction, $\sigma_Y(n)$. We observe a transition from IBE ($\sigma_Y(n)-\sigma_Y(p)>0$) to BE ($\sigma_Y(n)-\sigma_Y(p)<0$) as $\gamma_N$ increases beyond $\gamma_{Nc}$ at lower parent temperatures, $T_p \leq 0.10$. As the parent temperature increases, the peak height decreases and the IBE regime with $\sigma_Y(n)-\sigma_Y(p)>0$ shrinks, eventually giving way to an always BE regime with $\sigma_Y(n)-\sigma_Y(p)<0$ around $T_p \approx 0.12$. Panels (b) and (c) illustrate the evolution of yield stresses as a function of deformation history $\gamma_N$ at two parent temperatures, $T_p=0.035$ and $T_p=0.20$, measured in the positive (blue) and negative (red) shear directions; the insets show their difference, $\sigma_Y(n)-\sigma_Y(p)$, as in panel (a). The low-$T_p$ sample exhibits a clear inverse Bauschinger regime with a well-defined transition strain $\gamma_{Nc}$, whereas this regime is absent for the poorly annealed glass at $T_p=0.20$.
• Figure 5: Finite shear rate and directional memory. (a) The panel shows the difference between the yield stresses, $\sigma_Y(n)-\sigma_Y(p)$, as a function of the strain history $\gamma_N$. As the strain rate decreases, the peak height decreases and the IBE regime with $\sigma_Y(n)-\sigma_Y(p)>0$ shrinks, reducing $\gamma_{Nc}$ (=$\gamma_{N,\mathrm{crit}}(T_p,\dot{\gamma})$) and tending to zero around $\dot{\gamma} \approx 1\times10^{-4}$. Panels (b-e) show yield stresses as a function of deformation history $\gamma_N$ at $T_p=0.035$ for decreasing strain rates, $\dot{\gamma}=1\times10^{-3}$, $5\times10^{-4}$, $1\times10^{-4}$, and $5\times10^{-5}$, measured in the positive (blue) and negative (red) shear directions; the insets show their difference ($\sigma_Y(n)-\sigma_Y(p)$) as in (a). The higher strain rates $\dot{\gamma}=5\times10^{-3}$ (see Fig. \ref{['degree_of_Annealing']}(b)) and $\dot{\gamma}=1\times10^{-3}$ (panel (b)) exhibit a clear inverse Bauschinger regime with a well-defined transition, which becomes very small at $\dot{\gamma}=5\times10^{-4}$ and crosses over to the Bauschinger regime as $\gamma_N$ increases. At the lower strain rates $\dot{\gamma}=1\times10^{-4}$ and $5\times10^{-5}$, the inverse-Bauschinger regime is completely absent over the entire range of $\gamma_N$ studied.