On the anomalous elasticity in the mechanical response of amorphous solids

Abstract

The response of amorphous solids to a mechanical perturbation consists in an elastic and a plastic deformation. The latter is mediated by localized irreversible rearrangements associated with Eshelby-like quadrupolar singularities in the displacement field. It has recently been argued that a density of such singularities leads to an anomalous elastic behavior taking the form of screening effects, which goes beyond classical elastic predictions. Here, we reexamine this scenario using general theoretical arguments and a description in terms of an elasto-plastic model, which we compare with atomistic simulations of the canonical Eshelby inclusion geometry. We discuss the conditions under which a finite, i.e., nonvanishing, density of quadrupolar events is created by an imposed perturbation. We argue that, except when the perturbation is macroscopic, there are many situations in which the density of quadrupolar defects is zero in the thermodynamic limit. In these cases, we find that plastically active quadrupoles emerge in a region whose size generically scales as the spatial extent $\ell$ of the mechanical perturbation. This mechanism leads to anomalous elasticity on a scale $\ell$ close to the perturbation and to conventional elasticity beyond. The simulations of the elasto-plastic model reproduce the emergence of plastic quadrupoles in a region set by $\ell$ and the associated renormalization of the effective shear modulus, but they do not exhibit the dipole-screening signatures reported in atomistic and experimental studies. Our analysis delineates the scale-dependent breakdown of long-wavelength elasticity in amorphous materials and suggests directions for incorporating anomalous screening into mesoscopic modeling frameworks.

Figures (12)

• Figure 1: Sketch of the Eshelby problem: (a) In the atomic simulations of [hentschel_eshelby] a disk of radius $r_{\rm in}=5$ is instantaneously deformed into an ellipse of the same area with a deformation parameter $\delta=1.05$ in a sample bounded by an outer circle of radius $r_{\rm out}=80$ (the unit of length is the average atomic diameter). (b) In the present elasto-plastic model description, a large central plastic defects of diameter $\ell$ on the verge of yielding mimics the effect of an Eshelby inclusion in a system of linear size $L$ (with periodic boundary conditions).
• Figure 2: Elasto-plastic model: Stress-field difference induced by the initial Eshelby perturbation with a central defect of diameter $\ell=16$ and a system size $L=256$. Illustration for three different samples (left, middle, and right columns). The stress drop in the blocks forming the initial plastic defect is $\Delta\sigma_0=1$ and the shear modulus is taken as $\mu\equiv 1$. Top: full stress; middle: elastic component of the stress; bottom: plastic component of the stress. The initial plastic events associated with the central defect (shown as a blue core because the stress difference is $-\Delta \sigma_0=-1$) trigger additional plastic events (with aligned Eshelby-like quadrupoles) in the system. The color code is shown on the right of each panel.
• Figure 3: Elasto-plastic model: Stress-field difference induced by the initial Eshelby perturbation after averaging over 1000 independent samples for $\ell=4$ (left), $\ell= 8$ (middle), and $\ell = 16$ (right). The stress drop in the blocks forming the initial plastic defect (shown as a blue central core) is $\Delta\sigma_0=1$, $\mu=1$, and the system size is $L=256$. The color code is shown on the right of each panel.
• Figure 4: Atomic simulation: Sample-averaged XY component of the strain field for the Eshelby problem with $r_{\rm in}=5$, $r_{\rm out}=80$, and $\delta=1.05$ (see Fig. \ref{['fig_sketch-Eshelby']}(a)). Notice the overall $\cos(4\theta)$ symmetry (here, up to a phase). The XY strain field is obtained by combining simulation results for the displacement field in a poorly annealed glass and an analytical description of anomalous elasticity.hentschel_eshelbyavanish-eshelby (courtesy of Avanish Kumar and Itamar Procaccia).
• Figure 5: (a) Averaged number of plastically active sites $\overline N(r)$ in an annulus (centered on the origin) between the boundary of the central defect at $\ell/2$ and $r$ for several values of the initial defect size, $\ell=1$ (black), 2 (red), 4 (green), 8 (blue), 12 (light blue), and 16 (pink) (from bottom to top), and four different system sizes, $L=64$ (squares), $L=128$ (circles), $L=256$ (triangles), and $L=512$ (diamonds). The distance $r$ is normalized by the defect radius $\ell/2$. Notice the saturation toward a value $\overline{N}_{\rm sat}$ that is rather independent of the system size but grows with $\ell$. (b) The saturation value $\overline{N}_{\rm sat}$ grows as $\overline{N}_{\rm sat} \sim \ell^2$ (dashed line). (c) Averaged density of plastically active sites $\overline{N}(r)/(4\pi r^2)$ versus $r/(\ell/2)$.