Origin of the Bauschinger Effect in Amorphous Solids

We study the structural origin of the Bauschinger effect by accessing numerically the local plastic thresholds in the steady state flow of a two-dimensional model glass under athermal quasistatic deformation. More specifically, we compute the local residual strength, $\mathrm{\Delta }{\tau }^c$, for arbitrary loading orientations and find that plastic deformation generically induces material polarization, i.e., a forward-backward asymmetry in the $\mathrm{\Delta }{\tau }^c$ distribution. In steady plastic flow, local packings are on average closer to forward (rather than backward) instabilities, due to the stress-induced bias of barriers. However, presumably due to mechanical noise, a significant fraction of zones lie close to reverse (backward) yielding, as the distribution of $\mathrm{\Delta }{\tau }^c$ for reverse shearing extends quasilinearly down to zero local residual strength. By constructing an elementary model of the early plastic response, we then show that unloading causes reverse plasticity of a growing amplitude, i.e., reverse softening, while it shifts away forward-yielding barriers. This result in an inversion of polarization in the low-$\mathrm{\Delta }{\tau }^c$ region and, consequently, in the Bauschinger effect. This scenario is quite generic, which explains the pervasiveness of the effect.

Figure 1

Mean stress vs strain (both in absolute values) during three tests: unloading from steady flow (black), backward (red) and forward (green) loading from fully unloaded (zero stress) configurations. In all three cases, strain is measured with reference to the zero-stress state. Solid black line: fit of the unloading curve using Eq. (2) and $\rho a^2\mathrm{\Delta }{\epsilon }_0\simeq 0.25$. Solid green and red lines: consequent predictions for the Bauschinger tests. Dashed lines: predicted reloading and backward loading curves, when the model is used starting from the steady flow barrier distribution (see text), i.e., while taking into account its prediction for the unloading-induced asymmetry. Inset: corresponding evolution of mean barrier polarizations and their asymptotic values (dotted lines).

Figure 2

Left frames: the polar function $P(\mathrm{\Delta }{\tau }^c,2\alpha )$, with $⟨\mathrm{\Delta }{\tau }^c⟩(2\alpha )$ in white. (a) As-quenched isotropic (ESL) state, (b) steady state, and (c) unloaded steady state configurations. Right frames: cuts of $P(\mathrm{\Delta }{\tau }^c;2\alpha )$ along the forward ($2\alpha =0$) and backward ($2\alpha =\pi$) directions (d) in our three as-quenched glasses of different degrees of relaxation (HTL, ESL, and GQ) [25]. (e),(f) In steady flow (blue) and unloaded states (red), and (up to a scaling factor) model prediction for unloading (see text, dashed black).

Figure 3

Maps of the local residual strength for shearing (a) in the backward ($2\alpha =\pi$) and (b) forward ($2\alpha =0$) directions. Symbols show the loci of plastic events in the 2% of strain in the corresponding direction.