Plasticity-induced anisotropy in amorphous solids: The Bauschinger effect

Amorphous solids that underwent a strain in one direction such that they responded in a plastic manner “remember” that direction also when relaxed back to a state with zero mean stress. We address the question “what is the order parameter that is responsible for this memory?” and is therefore the reason for the different subsequent responses of the material to strains in different directions. We identify such an order parameter which is readily measurable, we discuss its trajectory along the stress-strain curve, and propose that it and its probability distribution function must form a necessary component of a theory of elastoplasticity.

Figure 1

(Color online) Stress-strain curves for a 2D system, see text. Left panel: starting the experiment from a freshly prepared sample results in a symmetric trajectory for $\gamma \to -\gamma$. Right panel: starting the experiment from the zero-stress state with $\gamma ={\gamma }_0$ results in an asymmetric trajectory, see text for details. Data were averaged over 500 independent stress-strain curves at $T=0.01$ where temperature is measured in units of $\epsilon /k_B$, see Sec. 2B below.

Figure 2

(Color online) Model stress-strain curves as obtained from $\sigma \sim {\sigma }_{\infty }\text{tanh}\left(\mu \left|\gamma -{\gamma }_0\right|/{\sigma }_{\infty }\right)+\beta {\left|\gamma -{\gamma }_0\right|}^2{e}^{-{\left|\gamma -{\gamma }_0\right|}^2}$ with $\beta$ positive.

Figure 3

(Color online) Upper panel: Trajectories of stress vs strain for a 2D system at four different temperature at the same strain rate $\dot{\gamma }={10}^{-4}$. Lower panel: the corresponding values of $⟨B_2⟩$ as a function of strain. Data were averaged over 1000 independent stress-strain curves at each temperature. Note that $⟨B_2⟩$ is negative even when the averaged stress-strain curve has a positive curvature, see text for discussion.

Figure 4

(Color online) The Bauschinger effect for the four temperatures shown in Fig. 3, increasing from top to bottom. The trajectories are displaced by fixed amount $\left(\Delta \left|{\sigma }_{xy}\right|=0.15\right)$ for clarity. Note the reduction of the effect with increasing temperature. Data were averaged over 500 independent stress-strain curves at each temperature. Inset: the shaded area of difference between the stress-strain curves with positive and negative strain as a function of $⟨B_2⟩$. The magnitude of the Bauschinger effect saturates for $T\to 0$.

Figure 5

(Color online) Probability distribution function of $B_2$ at various points on the stress-strain trajectory of a 2D system. Data were collected from 3000 independent stress-strain trajectories at $T=0.01$. In red (continuous) line we draw the symmetric pdf of the freshly prepared samples with $\gamma =0$. In green (dashed) line we show the pdf in the steady state, where it gains a negative asymmetry. In blue (dashed-dotted) line we see the pdf at the Bauschinger point $\gamma ={\gamma }_0$ where it gained a positive asymmetry. The dynamics of these pdf’s and their means are correlated with the shapes of the stress-strain curves and are proposed to be a crucial ingredient in any theory of elastoplasticity. Inset: the $N$ dependence of the pdf at the Bauschinger point $\gamma ={\gamma }_0$. Data was averaged over 1000 independent samples for each system size.

Figure 6

(Color online) A representative averaged stress-strain curve (averaged over 600 independent trajectories) for the three-dimensional model (upper panel) and the corresponding trajectory of $⟨B_2⟩$ in the lower panel for $T=0.01$.