Time-dependent elastic response to a local shear transformation in amorphous solids

The elastic response of a two-dimensional amorphous solid to induced local shear transformations, which mimic the elementary plastic events occurring in deformed glasses, is investigated via molecular-dynamics simulations. We show that for different spatial realizations of the transformation, despite relative fluctuations of order one, the long-time equilibrium response averages out to the prediction of the Eshelby inclusion problem for a continuum elastic medium. We characterize the effects of the underlying dynamics on the propagation of the elastic signal. A crossover from a propagative transmission in the case of weakly damped dynamics to a diffusive transmission for strong damping is evidenced. In the latter case, the full time-dependent elastic response is in agreement with the theoretical prediction, obtained by solving the diffusion equation for the displacement field in an elastic medium.

Figure 1

Sketch of a local shear transformation. Frozen particles inside the shear transformation region STR (dark gray particles), a circular region of radius $a$, are instantaneously displaced along the $x$ and $y$ directions according to the transformation defined in Eq. (3).

Figure 2

An example of the response to an induced local shear transformation. Snapshots of the vector displacement field were taken at different times, which feature the propagation of the elastic signal in the system.

Figure 3

Long-time mean displacement field $\langle {\mathbf{u}}_{\infty }\rangle$ after a local shear transformation (in the origin) obtained averaging over realizations of the transformation in different regions of the system. The quadrupolar structure is equivalent to one observed in isolated plastic events occurring in sheared glasses.

Figure 4

Main panel: symbols are the radial component of the long-time displacement field $\langle u_{r\infty }(r,\theta )\rangle$, along the $\theta =\pi /4$ direction, for different values of the damping $\tau$. Both isoconfigurational and disorder average are considered. Periodic boundary conditions are responsible for the field going to zero at the box boundaries. Full line is the prediction according to continuum elasticity theory, given by Eq. (5). Inset: fluctuations of the long-time radial displacement $\langle \delta u_{r\infty }(r,\theta )\rangle$ (symbols), compared to the mean displacement (lines). Relative fluctuations $\langle \delta u_{r\infty }\rangle /\langle u_{r\infty }\rangle$ are found of order one.

Figure 5

Time dependence of the total radial displacement ${\Delta }_r\left(t\right)$, defined in Eq. (15), in different conditions of damping. The transmission of the elastic response to a shear transformation changes from propagative, ${\Delta }_r\left(t\right)\propto t$, at short time to diffusive, ${\Delta }_r\left(t\right)\propto t^{1/2}$, at times longer than $\tau$.

Figure 6

Open symbols: $\langle u_r(r,\theta ,t)\rangle$, time-dependent radial component of the displacement field along the $\theta =\pi /4$ direction for highly damped dynamics, $\tau =0.1$ (top) and $\tau =0.01$ (bottom). Both isoconfigurational and disorder average are considered. Full lines: theoretical predictions according to Eqs. (13) and (14).