Nonaffine displacements and the nonlinear response of a strained amorphous solid

We demonstrate that irreversible structural reorganization is not necessary for the observation of yield behavior in an amorphous solid. While the majority of solids strained to their yield point do indeed undergo an irreversible reorganization, we find that a significant fraction of solids exhibits yield via a reversible strain. We also demonstrate that large instantaneous strains in excess of the yield stress can result in complete stress relaxation, a result of the large nonaffine motions driven by the applied strain. The empirical similarity of the dependence of the ratio of stress over strain on the nonaffine mean-square displacement to that for the shear modulus obtained from quiescent liquid at nonzero temperature supports the proposition that rigidity depends on the size of the sampled configurational space only and is insensitive to how this space is sampled.

Figure 1

Plot of the average change in stress $\langle \sigma \rangle$ vs $\gamma$ for the step strain and for the quasistatic strain. The yield strain ${\gamma }^{*}$ is indicated by the vertical dashed line.

Figure 2

Plot of the shear stress $\langle \sigma \rangle$ vs applied strain $\gamma$ calculated for the reversible strains (as explained in the text) (red squares) compared with the value from the all strains (black circles) The insert shows the probability $f_{\mathrm{rev}}\left(\gamma \right)$ of a strain being reversible as a function of the applied strain. The yield strain ${\gamma }^{*}$ is indicated by a vertical dashed line.

Figure 3

Plot of the nonaffine $\langle \mathrm{\Delta }{r}^2\rangle$ vs $\gamma$. The values of $\langle \mathrm{\Delta }{r}^2\rangle$ for the reversible and irreversible strains are shown separately, along with the overall value. The straight line is a fit corresponding to $\langle \mathrm{\Delta }{r}^2\rangle \propto {\gamma }^2$.

Figure 4

Plot of the reduced mechanical response $\frac{\langle \sigma \rangle }{G_{\infty }\gamma }$ vs $\langle \mathrm{\Delta }{r}^2\rangle$ for the $T=0$ solids subjected to strain (solid line). Also plotted is the reduced shear modulus $G_{\mathrm{eq}}/G_{\infty }$ from Ref. [16], calculated over a range of temperatures and averaging times in quiescent liquids. The infinite frequency modulus $G_{\infty }=\langle {\sigma }_{\mathrm{affine}}\rangle /\gamma$, calculated using a strain of ${10}^{-4}$.