Anomalous elasticity and plastic screening in amorphous solids

Amorphous solids appear to react elastically to small external strains, but in contrast to ideal elastic media, plastic responses abound immediately at any value of the strain. Such plastic responses are quasilocalized in nature, with the “cheapest” one being a quadrupolar source. The existence of such plastic responses results in screened elasticity in which strains and stresses can either quantitatively or qualitatively differ from the unscreened theory, depending on the specific screening mechanism. Here we offer a theory of such screening effects by plastic quadrupoles, dipoles, and monopoles, explain their natural appearance, and point out the analogy to electrostatic screening by electric charges and dipoles. For low density of quadrupoles the effect is to normalize the elastic moduli without a qualitative change compared to pure elasticity theory; for higher density of quadrupoles the screening effects result in qualitative changes. Predictions for the spatial dependence of displacement fields caused by local sources of strains are provided and compared to numerical simulations. We find that anomalous elasticity is richer than electrostatics in having a screening mode that does not appear in the electrostatic analog.

Figure 1

Displacement field induced by the inflation of one disk at the center of the box by 1%. The displacement vectors are normalized by the maximal one and plotted with the color code shown on the right. These are two different configurations prepared with identical protocols; see Appendix pp1 for details

Figure 2

Angle-averaged displacement field computed from the data presented in Fig. 1. Together with the data we present in continuous lines the theoretical predictions that are developed for the angle-averaged displacement in this paper. The parameters used in Eq. (45) to fit the data are the following: (a) $r_{\text{in}}=5,\kappa =0.0382,d_0=0.0368,r_{\text{out}}=97.18$; (b) $r_{\text{in}}=2.1,\kappa =0.0372,d_0=0.0847,r_{\text{out}}=98.37$.

Figure 3

Graphic representation of the solution of Eq. (9) for $r_{\text{in}}=0.01$ and $r_{\text{out}}=1$, with $d_r\left(r_{\text{in}}\right)=0.1$ and $d_r\left(r_{\text{out}}\right)=0$.

Figure 4

Displacement field for $P=40$. Inset: the displacement field caused by inflating one disk closest to the origin by 1%. Here the displacement vectors are normalized and color coded as in Fig. 1. Shown also is the fit of Eq. (9) to the angle-averaged displacement. Here the parameters used in the fit are $d_0=0.03028, r_{\text{in}}=3.268, r_{\text{out}}=82.74$.

Figure 5

Displacement field for $P=60$. Inset: the displacement field caused by inflating one disk closest to the origin by 1%. The displacement vectors are normalized and color coded as in Fig. 1 Shown also is the fit of Eq. (9) to the angle-averaged displacement. Here the parameters used in the fit are $d_0=0.03487, r_{\text{in}}=3.631, r_{\text{out}}=81.9$. We note that $r_{\text{in}}$ is of the order of the inclusion, and $r_{\text{out}}$ of the system radius.

Figure 6

The solutions of Eq. (45) for different values of the parameter $\kappa$. Here $r_{\text{in}}=0.01$ and $r_{\text{out}}=1$, with $d_r\left(r_{\text{in}}\right)=0.1$ and $d_r\left(r_{\text{out}}\right)=0$.

Figure 7

Comparison of typical displacement fields with respect to the unperturbed configuration, $P=60$. Upper panel: after inflation. Lower panel: after deflation.

Figure 8

Comparison of typical displacement fields with respect to the unperturbed configuration, $P=4.5$. Upper panel: after inflation. Lower panel: after deflation.