Atomistic simulation study of the shear-band deformation mechanism in Mg-Cu metallic glasses

We have simulated plastic deformation of a model Mg-Cu metallic glass in order to study shear banding. In uniaxial tension, we find a necking instability occurs rather than shear banding. We can force the latter to occur by deforming in plane strain, forbidding the change of length in one of the transverse directions. Furthermore, in most of the simulations a notch is used to initiate shear bands, which lie at a 45° angle to the tensile loading direction. The shear bands are characterized by the Falk and Langer local measure of plastic deformation $D_{\min }^2$, averaged here over volumes containing many atoms. The $D_{\min }^2$ profile has a peak whose width is around $10\mathrm{nm}$; this width is largely independent of the strain rate. Most of the simulations were, at least nominally, at $100\mathrm{K}$, about $T_g∕3$ for this system. The development of the shear bands takes a few tens of ps, once plastic flow has started, more or less independent of strain rate. The shear bands can also be characterized using a correlation function defined in terms of $D_{\min }^2$, which, moreover, can detect incipient shear bands in cases where they do not fully form. By averaging the kinetic energy over small regions, the local temperature can be calculated, and this is seen to be higher in the shear bands by about $50–100\mathrm{K}$. Increases in temperature appear to initiate from interactions of the shear bands with the free surfaces and with each other, and are delayed somewhat with respect to the localization of plastic flow itself. We observe a slight decrease in density, up to 1%, within the shear band, which is consistent with notions of increased free volume or disorder within a plastically deforming amorphous material.

Figure 1

Schematic diagrams of the main geometries used. B and D have notches while A and C do not. The large arrows indicate the direction of applied strain.

Figure 2

(Color) Stress-strain curves for samples A, B, C, and D deformed at the faster strain rate.

Figure 3

(Color) Outline of sample A deformed in uniaxial tension after 0%, 20%, 40%, 60%, 80%, and 100% strain.

Figure 4

(Color) Minimum and maximum widths in the $y$ direction, for samples A (black) and B (red) deformed in uniaxial tension, as a function of strain.

Figure 5

Visualization by stripe painting in sample D deformed at the faster strain rate, at different amounts of strain.

Figure 6

Shear bands visualized using $D_{\mathit{min}}^2$ in sample D deformed at the faster strain rate, at (a) 5%, (b) 10%, (c) 20%, and (d) 30% strain.

Figure 7

Profiles of $D_{\mathit{ave}}^2$ from samples (a) D at 5%, (b) D at 10%, (c) E at 5%, (d) E at 10% strain, deformed at the “faster” strain rate. Solid, dotted, and dashed lines are for $y^{\prime }$ centered on $-10$, $-20$, and $-30\mathrm{nm}$, respectively, for sample D. Solid, dotted lines are for $y^{\prime }$ centered on $-9$ and $-13\mathrm{nm}$, respectively, for sample E (note here that the data fall to zero at negative $x^{\prime }$ corresponding to the edge of the sample).

Figure 8

(Color) Stress-strain curves compared for different strain rates, and with/without thermostat, for sample D. The inset shows the temperature, determined from the average kinetic energy per atom, as a function of strain for the different cases.

Figure 9

Visualization using $D_{\mathit{min}}^2$ in sample D deformed at the slower strain rate, at 4%, 5%, 10%, and 15% strain. At this strain rate the shear bands are already well developed at 5% strain. They also appear to be more intense, particularly the one on the right.

Figure 10

Lattice correlation function $C_D$ from sample D, at the faster strain rate. Left, averaged between 5% and 10% strain; right, averaged between 10% and 15% strain.

Figure 11

Lattice correlation function for $D_{\mathit{ave}}^2$ from sample D, the slower strain rate, at 5% and 10% strain. It is clear that the shear band on the right of the sample, parallel to the line $y=-x$, is significantly more intense.

Figure 12

Above, the lattice correlation function for $D_{\mathit{ave}}^2$ from sample C, the faster strain rate, averaged between 25% and 30% strain. Below, visualization using $D_{\mathit{min}}^2$ at 25% strain. The emergence of shear bands parallel to $y=x$ is clear in the first case and discernible in the second.

Figure 13

Above, the visualization of $D_{\mathit{min}}^2$ in sample C deformed at the slower strain rate at (a) 5% (b) 10%, and (c) 15% strain. Below, the correlation function averaged over 5—7.5%, 10—12.5%, and 15—17.5%. There is clearly an evolution toward a state with well-defined shear bands.

Figure 14

(Color) Temperature plot of sample D deformed at the faster strain rate at $100\mathrm{K}$, at 20% strain. Each square is $2.5\mathrm{nm}\times 2.5\mathrm{nm}$. The temperature is seen to be increased around the shear bands.

Figure 15

(Color) Temperature plot of sample D deformed at the slower strain rate at $100\mathrm{K}$, at 30% strain.

Figure 16

(Color) Temperature plot of a sample D deformed at the faster strain rate with thermostat turned off after equilibration to $100\mathrm{K}$, 20% strain.

Figure 17

(Color) Peak heights of profiles of $D_{\mathit{ave}}^2$ (black) and temperature (red) for sample D deformed at the faster strain rate. Solid and dotted lines denote profiles at $y^{\prime }=-20\mathrm{nm}$ and $-30\mathrm{nm}$, respectively. Inset: the same for the slower strain rate, $y^{\prime }=-20\mathrm{nm}$. It is seen that $D_{\mathit{ave}}^2$ rises first, and then temperature.

Figure 18

(Color) Temperature plot of sample G deformed at the faster strain rate at $100\mathrm{K}$, 25% strain. The temperature rise is seen to be larger, where the shear bands meet the surface, particularly on the left.

Figure 19

(Color) Temperature plot of sample H deformed at the faster strain rate at 35% strain. The temperature rise is increased where the shear bands intersect.

Figure 20

(Color) Density plots of sample D deformed at strain rate S. The coarse-graining length is $2.5\mathrm{nm}$ in the upper plots and $7.5\mathrm{nm}$ in the lower ones. The plots on the left are for the undeformed system, while those on the right are at 20% strain. Dark squares at the edge are typically boxes that are only partially filled with atoms. A reduction of density in the areas associated with shear bands is apparent in the right figures.