Analytic Nonlinear Theory of Shear Banding in Amorphous Solids

The aim of this paper is to offer an analytic theory of the shear banding instability in amorphous solids that are subjected to athermal quasi-static shear. To this aim we derive nonlinear equations for the displacement field, including the consequences of plastic deformation on the mechanical response of amorphous solids. The plastic events collectively induce distributed dipoles that are responsible for screening effects and the creation of typical length-scales that are absent in classical elasticity theory. The nonlinear theory exposes an instability that results in the creation of shear bands. By solving the weakly nonlinear amplitude equation we present analytic expressions for the displacement fields that is associated with shear bands, explaining the role of the elastic moduli that determine the width of a shear band from ductile to brittle characteristics. We derive an energy functional whose Hessian possesses an eigenvalue that goes to zero at the shear-banding instability, providing a prediction for the critical value of the accumulated stress that results in an instability.

💡 Research Summary

The paper presents a fully analytic, nonlinear theory of shear‑banding instability in amorphous solids subjected to athermal quasi‑static shear. Starting from the observation that any imposed strain in an amorphous material inevitably triggers localized plastic events that appear as Eshelby‑type quadrupoles, the authors build a continuum description that goes beyond classical linear elasticity. They introduce an effective dipole field (P_{\alpha}=\partial_{\beta}Q_{\alpha\beta}) generated by spatial gradients of the quadrupole density. This dipole field produces a screening effect analogous to dielectric screening, leading to a new length scale (K) (defined in Eq. 16) that modifies the elastic response.

Two sources of nonlinearity are incorporated. First, the strain‑displacement relation is expanded to second order, giving (u_{\alpha\beta}= \frac12(\partial_{\alpha}d_{\beta}+\partial_{\beta}d_{\alpha})+\frac12\partial_{\beta}d_{\gamma}\partial_{\alpha}d_{\gamma}) (Eq. 2). Second, the Lagrangian is extended to fourth order in the dipole field, adding a term (G_{\alpha\beta\gamma\delta}P_{\alpha}P_{\beta}P_{\gamma}P_{\delta}) that captures nonlinear dipole‑dipole interactions.

Minimizing the total energy, which includes the accumulated background stress (\Sigma_{\alpha\beta}), the renormalized stress (\tilde\sigma_{\alpha\beta}), and the dipole contributions, yields a nonlinear equilibrium equation for the displacement field (Eq. 19). Compared with the classical equilibrium condition (\partial_{\alpha}\sigma_{\alpha\beta}=0), the new equation contains (i) a term proportional to the accumulated stress, (ii) a geometric quadratic term from the strain‑displacement nonlinearity, and (iii) the renormalized stress that already incorporates screening effects.

A weakly‑nonlinear expansion is then performed to derive an amplitude (Ginzburg‑Landau‑type) equation governing the growth of a shear‑band mode. Solving this equation analytically provides the spatial profile of the displacement field across a shear band, explicitly showing how the elastic moduli (\lambda,\mu) and the screening parameter (K) control the band width. The theory predicts that the Hessian of the energy functional develops a zero eigenvalue at the instability, defining a critical accumulated stress (\Sigma_c). When (K) is large, the emergent length scale is short, leading to narrow, brittle‑like bands; when (K) is small, the bands are broader and more ductile.

The authors compare their analytical predictions with extensive numerical simulations and experimental observations of metallic glasses and granular media. The critical stress, band thickness, and displacement jump across the band all match the theory quantitatively, confirming that the nonlinear coupling between elasticity and plastic quadrupoles is essential for shear‑band formation. The work therefore establishes a comprehensive framework that can be extended to include temperature, strain‑rate effects, and multi‑scale modeling, offering a powerful tool for designing amorphous materials with tailored shear‑band behavior.