Elastic interactions of plastic events in strained amorphous solids before yield

It has been widely accepted that the plastic events of amorphous solids after mechanical yield belong to a highly correlated avalanche state. However, whether the plastic events before yield are correlated or not is still unsettled, leaving their interactions largely unexplored. In this paper, by means of atomistic simulations, typical ${\mathrm{Cu}}_{50}{\mathrm{Zr}}_{50}$ metallic glasses, as the model system, are sheared under athermal quasistatic limit to study these plastic events. The statistical analysis of both stress drops and waiting times reveals that plastic events before yield are in the correlated avalanche state and the interactions among them are mediated by the robust elasticity. The temporal correlation analysis of the nonaffine displacement fields further reveals that the elastic interactions are short-lived strong but long-standing weak, which results in the fractal morphology of potential energy landscape. By introducing vibrational modes to explore plastic events, we clearly exhibit the way how the elastic interactions organize the Eshelby-type shear transformations into avalanched plastic events. The correlation matrix, with its component being the dot product of the vibrational modes at different configurations, is defined to trace the evolution of vibrational modes during elastic deformation and across plastic events. Three reasons accounting for the robust elasticity are identified: (i) the limited destruction of plastic events on global elasticity, (ii) the persistent hard spots embedded in elastic matrix, and (iii) the self-recovery of elastic matrix during elastic deformation. Our results clarify the atomic-scale nature of both elastic deformation and plastic instabilities before yield in amorphous solids, providing fundamental information for the development of elastoplastic constitutive models.

Figure 1

Atomic model of ${\mathrm{Cu}}_{50}{\mathrm{Zr}}_{50}$ metallic glass: (a) the 3D packing configuration; (b) the radial distribution function.

Figure 2

(a) Thirty stress-strain curves of five independent samples sheared in six different directions. The black thick one belongs to the selected representative sample. (b) A partial magnification of the black thick curve in (a). The curve consists of plastic events and elastic portions, which are quantitatively characterized by stress drops and waiting times, respectively.

Figure 3

(a) Stress drops and (b) normalized stress drops in 30 stress-strain curves as a function of the applied strain. The vertical dashed line in (b) indicates the upper limit of the strain range 0–0.1, within which plastic events are studied.

Figure 4

Probability density function of normalized stress drops at different deformation stages shown by legend. The power-law exponents changes from about 1 in strain range 0–0.02 to about 1.5 in strain range 0.02–0.1 with mechanical loading.

Figure 5

Relationship between $DPN$ and $\parallel \mathrm{\Delta }{\mathbf{R}}_{\mathrm{plastic}}\parallel$. $DPN$ increases with $\parallel \mathrm{\Delta }{\mathbf{R}}_{\mathrm{plastic}}\parallel$ and has a wide range from several to several hundreds.

Figure 6

(a) The 3D spatial distribution of the cumulative atomic shear strain ${\gamma }_{xy}$ within strain range 0–0.1. Sample is sheared along $xy$ direction. (b) Plot of the $x-y$ cross section of the 3D spatial autocorrelation function of the cumulative atomic shear strain ${\gamma }_{xy}$ in (a), exhibiting a characteristic quadrupole pattern.

Figure 7

(a) Waiting times in 30 stress-strain curves as a function of the applied strain. (b) Probability density function of waiting times shows an obvious peak close to 25 strain steps.

Figure 8

Correlation function of nonaffine displacement fields as a function of strain interval. There are four categories of curves shown by legend according to the properties of nonaffine displacement fields.

Figure 9

Nonaffine displacement matrix reflects fractal characteristics of PEL. (a) Nonaffine displacement matrix within the strain range 0–0.1. (b), (d) Magnification of two regions represented by dashed window in (a). (c) and (e) correspond to (b) and (d) with a more precise color scale, respectively.

Figure 10

Participation degree of the vibrational modes of three scale plastic events. (a)–(c) correspond to small-, medium-, and large-scale events, whose $\parallel \mathrm{\Delta }{\mathbf{R}}_{\mathrm{plastic}}\parallel$ is 2.6, 10.7, and 47.5 $\AA ·{\left(\mathrm{g}/\mathrm{mol}\right)}^{1/2}$, respectively. The green circles in (a)–(c) correspond to the lowest-frequency mode ${\mathit{\Psi }}_1$ and the orange triangle in (b) corresponds to the second-lowest frequency mode ${\mathit{\Psi }}_2$.

Figure 11

Nonaffine displacement field $\mathrm{\Delta }{\mathbf{R}}_{\mathrm{plastic}}$ of the small-scale event and its dominant vibrational mode. (a) $\mathrm{\Delta }{\mathbf{R}}_{\mathrm{plastic}}$ of this event in 3D real space. Vector field in (b) is the slice of (a) in $xy$ plane and the radii of green circles are proportional to the magnitude of the polarization vectors of the lowest-frequency mode ${\mathit{\Psi }}_1$.

Figure 12

Nonaffine displacement field $\mathrm{\Delta }{\mathbf{R}}_{\mathrm{plastic}}$ of the medium-scale event and its dominant vibrational modes. (a) Plot of the total potential energy as a function of the minimization step during the energy minimization process. Two STs corresponding to the abrupt energy drops can be clearly identified. (b) $\mathrm{\Delta }{\mathbf{R}}_{\mathrm{plastic}}$ of this event in 3D real space. Vector fields in (c)–(e) correspond to the $xy$-plane slice of $\mathrm{\Delta }{\mathbf{R}}_{\mathrm{plastic}}$ of this event, the first ST, and the second ST, respectively. Radii of green circles in (c) and (d) are proportional to the magnitude of the polarization vectors of the lowest-frequency mode ${\mathit{\Psi }}_1$, and the radii of orange circles in (c) and (e) are proportional to the magnitude of the polarization vectors of the second-lowest frequency mode ${\mathit{\Psi }}_2$.

Figure 13

Nonaffine displacement field $\mathrm{\Delta }{\mathbf{R}}_{\mathrm{plastic}}$ of the large-scale event and its dominant vibrational modes. (a) $\mathrm{\Delta }{\mathbf{R}}_{\mathrm{plastic}}$ of this event in 3D real space. Vector field in (b) is the slice of (a) in $xy$ plane. Radii of green circles are proportional to the magnitude of the polarization vectors of the lowest-frequency mode ${\mathit{\Psi }}_1$, which has the largest participation degree. And, the radii of pink, orange, cyan, and blue circles are, respectively, proportional to the magnitude of the polarization vectors of the vibrational modes ${\mathit{\Psi }}_7, {\mathit{\Psi }}_4, {\mathit{\Psi }}_{10}$, and ${\mathit{\Psi }}_{23}$, which have the second-, third-, fourth-, and fifth-largest participation degree. Vibrational modes are numbered in ascending order of frequency.

Figure 14

Relationship between $P{D}_{\max }$ and $\parallel \mathrm{\Delta }{\mathbf{R}}_{\mathrm{plastic}}\parallel$. Red solid arrow is drawn as a guide for the eyes.

Figure 15

Dependence of identification ratio on $\parallel \mathrm{\Delta }\mathbf{R}\parallel$ for elastic portions and plastic events.

Figure 16

Comparison of $ECC$ between plastic event (a) and elastic portion (b) under equal $\parallel \mathrm{\Delta }\mathbf{R}\parallel$ of 8.3 $\AA ·{\left(\mathrm{g}/\mathrm{mol}\right)}^{1/2}$. Green circle in (a) denotes $ECC$ of the critical mode.

Figure 17

Comparison of the frequency variation of identifiable modes between plastic event and elastic portion with equal $\parallel \mathrm{\Delta }\mathbf{R}\parallel$ of 8.3 $\AA ·{\left(\mathrm{g}/\mathrm{mol}\right)}^{1/2}$; the chosen cases are the same as that in Figs. 16 and 16 and correspond to the increase and decrease of the frequencies of plastic event, respectively. (b) and (d) correspond to the increase and decrease of the frequencies of elastic portion, respectively.