Analytic theory of shear localization in amorphous solids confined by Couette geometry

TL;DR

The paper develops an analytic theory for shear localization in a quasi-statically loaded 2D amorphous solid under Couette confinement, showing that plastic events act as topological quadrupoles and dipoles which renormalize the elastic response. The authors derive a linear displacement equation $\Delta \mathbf{d}+(1+\tilde{\lambda})\nabla(\nabla\cdot\mathbf{d})+\boldsymbol{\Gamma}\mathbf{d}=\mathbf{0}$ with a discrete screening parameter $\kappa_e$ governing the solution, and obtain explicit Bessel-function forms for the angle-averaged tangential displacement $d_\theta(r)$ under boundary conditions $d_r(R_{in})=d_r(R_{out})=0$, $d_\theta(R_{in})=\Omega_0$, $d_\theta(R_{out})=0$. The theory predicts a priori where shear localization occurs (near inner boundary or in the bulk) by selecting discrete $\kappa_e$ values, which are shown to correspond to observed displacements in simulations. The work unifies plastic-event topology with continuum elasticity to explain ductile shear localization and provides a framework to predict displacement fields from stress drops in quasi-static loading, with potential extensions toward nonlinear effects that could describe shear-bands in brittle materials.

Abstract

``Couette geometry'' refers to two concentric rings in 2-dimensions (or cylinders in 3-dimensions with a medium in between. Typically the inner and outer rings (or cylinders) rotate at different rates and the response of the medium is studied. Here we study a medium which is a twodimensional amorphous solid, and we rotate the inner ring quasi-statically. As stress accumulates, plastic avalanches can result in shear localization, characterized by adjacent parts of the system rotating in opposite directions, with the maximum shear localized between them. We derive an analytic theory that describes and explains the shear localization, providing a-priori predictions for the displacement field associated with the plastic drops and the shear localization.

Figures (5)

• Figure 1: Panel (a): The simulated system, here the area fraction is $\phi=0.91$, see text for details. Panel (b): typical shear stress vs. accumulated rotation angle $\Theta$. The stress drop responsible for Panel (c) is indicated by an arrow. Panel (c): typical profile of the angle-averaged angular component of the displacement field $d_\theta(r)$ normalized by inner rotation $\Omega_0$ as a function of $r$, showing the reversal of particle displacement from anti-clockwise to clockwise. This leads to the stress localization shown in Fig. \ref{['loc']}. The blue curve represents the tangential displacement of the elastic solution shown in Eq. (\ref{['d2']}).
• Figure 2: The shear strain as a function of $r$, exhibiting the shear localization at $r=51.4$ where the displacement field in Panel (c) of Fig. \ref{['Fig1']} changes sign. The blue curve represents the shear strain of the elastic solution.
• Figure 3: Plot of the maximal value of $|d_\theta(r)|$ normalized by inner rotation $\Omega_0$ as a function of the screening parameter $\kappa_e$. The peaks represent the values of $\kappa_e$ where the denominator of Eq. (\ref{['solbess']}) goes to zero, and the response of the system is maximal. The blue dots represent best fits of $\kappa_e$ to actual measured displacement fields in our simulations. The preference of the system to select responses characterized by discrete values of $\kappa_e$ is obvious. The height of every blue dot represents the stress drop $-\sigma^{r\theta}$ that is defined as the difference in total stress between the mechanical equilibrium configurations before and after a small rotation at the inner boundary which resulted in the fitted displacement field.
• Figure 4: Color maps of the displacement field. Red indicates anti-clockwise and blue clockwise displacement. Panel (a): Typical map of displacement field when the screening exponent $\kappa_e$ is in the vicinity of $\kappa_e =0.067$. In this case the shear localization occurs near the inner boundary. Panel (b): Typical map of displacement field when the screening exponent $\kappa_e$ is in the vicinity of $\kappa_e =0.123$. Here the shear localization occurs in the bulk.
• Figure S1: Other examples of the profile of the angle-averaged angular component of the displacement field $d_\theta(r)$ normalized by inner rotation $\Omega_0$ as a function of $r$, showing the reversal of particle displacement from anticlockwise to clockwise where the shear localization occurs near the boundary (Panel (a-b)) and in the bulk (Panel (c-d)). The stress drops from Panel (a-d) correspond to 4.77, 19.8, 9.35, and 7.85, respectively, and fitted $\kappa_e$ corresponding to 0.067, 0.066, 0.111, and 0.109, respectively.