Mechanical properties and plasticity of a model glass loaded under stress control

Much of the progress achieved in understanding plasticity and failure in amorphous solids had been achieved using experiments and simulations in which the materials were loaded using strain control. There is paucity of results under stress control. Here we present a method that was carefully geared to allow loading under stress control either at $T=0$ or at any other temperature, using Monte Carlo techniques. The method is applied to a model-perfect crystalline solid, to a crystalline solid contaminated with topological defects, and to a generic glass. The highest yield stress belongs to the crystal, the lowest to the crystal with a few defects, with the glass in between. Although the glass is more disordered than the crystal with a few defects, it yields stress much higher than that of the latter. We explain this fact by considering the actual microscopic interactions that are typical of glass-forming materials, pointing out the reasons for the higher cohesive nature of the glass. The main conclusion of this paper is that the instabilities encountered in stress-control condition are the identical saddle-node bifurcation seen in strain control. Accordingly one can use the latter condition to infer the former. Finally we discuss temperature effects and comment on the time needed to see a stress-controlled material failure.

Figure 1

Interaction potentials in dimensionless units.

Figure 2

Configurations of the one-component system. Top panel: Configuration I with perfect hexagonal structure. Bottom panel: Configuration II with defects. The dotted line represents the simulation cell which is continued periodically in both directions.

Figure 3

Distributions of the stress tensor components (upper panel) and strain tensor components (bottom panel) of the perfect hexagonal structure at ${\sigma }^{\mathrm{ext}}=0$ at temperature $T=0.05$ ($\alpha =x,y$).

Figure 4

Shape of the simulation box. The segment AE is perpendicular to the side CD.

Figure 5

Stress-strain dependence under stress control for system I (see Fig. 2). The blue dots represent results of the Monte Carlo simulations at $T=0.05$. The red line represents the prediction of the athermal quasistatic protocol (at $T=0$). Note that at zero temperature the yield stress is considerably larger.

Figure 6

The dependence of the internal stress (top panel) and the energy (bottom panel) on the strain under stress control for system I (see Fig. 2).

Figure 7

The strain dependence of the generalized enthalpy in the athermal case for system I. The input from the strain-controlled experiment is the first curve at ${\sigma }_{xy}^{\mathrm{ext}}=0$. To this function we now add the term $-V{\sigma }^{\mathrm{ext}}{\gamma }_r$ according to Eq. (18) to get all the other curves at varying values of ${\sigma }_{xy}^{\mathrm{ext}}$.

Figure 8

The evolution of the energy (top panel) and the strain $\gamma$ (bottom panel) during the MC protocol for system II (see Fig. 2) for two values of the temperature. The external stress ${\mathit{\sigma }}_{xy}^{\mathrm{ext}}=0$, $T=0.05$ (black line), and $T=0.1$ (gray line).

Figure 9

Evolution of the strain $\gamma$ in the MC simulations under applied external stress.

Figure 10

Dependence of the internal stress (top panel) and the energy (bottom panel) on the strain under stress control for system II (see Fig. 2). Colors correspond to the legend in Fig. 9.

Figure 11

The strain dependence of the generalized enthalpy in the athermal case for system II. The input from the strain-controlled experiment is the first curve at ${\sigma }_{xy}^{\mathrm{ext}}=0$. To this function we now add the term $-V{\sigma }^{\mathrm{ext}}{\gamma }_r$ according to Eq. (18) to get all the other curves at varying values of ${\sigma }_{xy}^{\mathrm{ext}}$.

Figure 12

Simulation results for system II (see Fig. 2). Upper panel: Distribution of strain values for a given external stress in the Monte Carlo simulation at $T=0.05$. For zero external stress there are only two available states corresponding to the two maxima in the distribution of the strain, or to the two minima in the free energy. For higher values of the external stress there are more available states which appear as additional peaks in the distribution of strain. The measured strain in the lower panel is computed as an average over the distributions in the upper panel. Lower panel: stress-strain dependence under stress control (blue dots) and for AQS (red line). The black triangle represents the state obtained by an MC simulation at ${\sigma }_{xy}^{\mathrm{ext}}=0$ starting from an initial condition which is a configuration found by the Monte Carlo protocol at ${\sigma }_{xy}^{\mathrm{ext}}=0.01$.

Figure 13

The structure of the binary mixture. Blue circles correspond to A particles, black circles to particles B.

Figure 14

Dependence of the internal stress (top panel) and the energy (bottom panel) on strain under stress control for the glass model (see Fig. 13).

Figure 15

Top panel: The strain dependence of the generalized enthalpy in the athermal case for the glass. The input from the strain-controlled experiment is the first curve at ${\sigma }_{xy}^{\mathrm{ext}}=0$. To this function we now add the term $-V{\sigma }^{\mathrm{ext}}\gamma$ according to Eq. (17) to get all the other curves at varying values of ${\sigma }_{xy}^{\mathrm{ext}}$. Bottom panel: Stress vs strain in a strain-controlled simulation of the response of the binary glass at AQS conditions.

Figure 16

Stress-strain dependence of the binary glass. We again stress that in the stress-control simulation the external stress is fixed and shown are the results for varying this fixed stress. In a triangle we show the state obtained by the MC simulation at ${\sigma }_{xy}^{\mathrm{ext}}=0$ with initial configuration from the run at ${\sigma }_{xy}^{\mathrm{ext}}=0.15$.