Local yield stress statistics in model amorphous solids

We develop and extend a method presented by Patinet, Vandembroucq, and Falk [Phys. Rev. Lett. 117, 045501 (2016)] to compute the local yield stresses at the atomic scale in model two-dimensional Lennard-Jones glasses produced via differing quench protocols. This technique allows us to sample the plastic rearrangements in a nonperturbative manner for different loading directions on a well-controlled length scale. Plastic activity upon shearing correlates strongly with the locations of low yield stresses in the quenched states. This correlation is higher in more structurally relaxed systems. The distribution of local yield stresses is also shown to strongly depend on the quench protocol: the more relaxed the glass, the higher the local plastic thresholds. Analysis of the magnitude of local plastic relaxations reveals that stress drops follow exponential distributions, justifying the hypothesis of an average characteristic amplitude often conjectured in mesoscopic or continuum models. The amplitude of the local plastic rearrangements increases on average with the yield stress, regardless of the system preparation. The local yield stress varies with the shear orientation tested and strongly correlates with the plastic rearrangement locations when the system is sheared correspondingly. It is thus argued that plastic rearrangements are the consequence of shear transformation zones encoded in the glass structure that possess weak slip planes along different orientations. Finally, we justify the length scale employed in this work and extract the yield threshold statistics as a function of the size of the probing zones. This method makes it possible to derive physically grounded models of plasticity for amorphous materials by directly revealing the relevant details of the shear transformation zones that mediate this process.

Figure 1

(a) Typical stress-strain curves for the three quench protocols: instantaneous quench from a high-temperature liquid (HTL, continuous line), instantaneous quench from an equilibrated supercooled liquid (ESL, dashed line), and a gradual quench (GQ, dash-dotted line). The inset shows a zoom of a stress drop corresponding to one plastic event. (b) Plastic event computed between the onset of instability and just after the event: the arrows and the color scale are the displacement $\vec{u}$ and maximum shear strain ${\eta }_{\text{VM}}$ fields, respectively. For the sake of clarity, the arrows are magnified by a factor 200 and deleted in the core region. The atom with the maximum shear strain gives the location of the plastic rearrangement.

Figure 2

(a) Schematic drawing of the local yield stress computation on a regular square grid of mesh size $R_{\text{sampling}}$. Region I (of radius $R_{\text{free}}$) is fully relaxed while region II (of width $2R_{\text{cut}}$) is forced to deform following an affine pure shear deformation in the $\alpha$ direction. (b) Typical local stress increment-strain curves of the Region I for different loading directions $\alpha =0,45,90$, and $-{45}^{\circ }$. The measurement of the threshold $\mathrm{\Delta }{\tau }_c$ and relaxation $\mathrm{\Delta }{\tau }_r$ are represented for $\alpha ={90}^{\circ }$. Inset: zoom one the low strain region of the corresponding stress-strain curves, i.e., without subtracting the initial local shear stress within the as-quenched glass ${\tau }_0\left(\alpha \right)$.

Figure 3

Local yield stress maps computed on a regular grid for the three different quench protocols: (a) HTL, (b) ESL, and (c) GQ. The first ten plastic event locations are shown as open black symbols numbered by order of appearance during remote shear loading. Plastic events clearly tend to occur in regions characterized by low-yield stresses.

Figure 4

Probability distribution function of the local yield stresses for the three different quench protocols. The corresponding cumulative distribution functions are represented in log-log scale in the inset. The straight line is a power law of exponent $1+\theta =1.6$, i.e., the expected scaling of the integral of the probability distribution function as $\mathrm{\Delta }{\tau }_y$ approaches zero.

Figure 5

Correlation between the local yield stresses computed in the quenched state and the locations of the plastic rearrangement as a function of the applied strain for the three quench protocols. The error bars correspond to one standard deviation. (a) Correlation computed from individual plastic rearrangements using Eq. (5). The arrows correspond to the average strain of the tenth plastic event. The lines are empirical $A+B{e}^{-{(\gamma /{\gamma }_d)}^2}$ fits from which the decorrelation strain ${\gamma }_d$ is estimated. (b) Pearson correlation coefficient computed between the local yield stress fields and the local strain fields using Eq. (6). Error bars are smaller than symbols.

Figure 6

(a) Probability distribution function of the stress drops for the three different quench protocols in lin-log scale. The same quantities rescaled by the shear moduli, i.e., the slip increments, are represented in the inset. The lines are exponential fits. (b) Average stress drop as a function of the average local yield stress for the free different quench protocols. The line is an exponential fit.

Figure 7

Top row: local yield stress maps computed on a regular grid for different loading directions for the GQ protocol: (a) simple shear ${\alpha }_l=0^{\circ }$, (b) pure shear ${\alpha }_l={45}^{\circ }$, and (c) negative simple shear ${\alpha }_l={90}^{\circ }$. The red arrows correspond to the applied strains. The first ten plastic event locations are shown as open black symbols numbered by order of appearance during remote shear loading. Bottom row: local yield stress contrasts $\text{TC}({\alpha }_l^1,{\alpha }_l^2)$ defined in Eq. (8) between the above loading directions: (d) ${\alpha }_l^1=0^{\circ },{\alpha }_l^2={45}^{\circ }$, (e) ${\alpha }_l^1={45}^{\circ },{\alpha }_l^2={90}^{\circ }$, and (f) ${\alpha }_l^1=0^{\circ },{\alpha }_l^2={90}^{\circ }$.

Figure 8

Cross-correlation of the local yield stress field as a function of the loading direction shift. Error bars are smaller than symbols.

Figure 9

Probability distribution functions of angles for the three quench protocols for which the projected local yield stress is minimal ${\alpha }_{\text{min}}$. The lines correspond to Gaussian fits which standard deviations are reported in the inset.

Figure 10

Same as Fig. 5 for the GQ protocol for different local yield stress fields: its minimum over all orientations ${min}_{\alpha }\mathrm{\Delta }{\tau }_c\left(\alpha \right)$, its value along the loading direction $\mathrm{\Delta }{\tau }_c(\alpha ={\alpha }_l)$ and its minimum once projected along the loading direction ${min}_{\alpha }\mathrm{\Delta }{\tau }_c\left(\alpha \right)/cos\left[2(\alpha -{\alpha }_l)\right]$ [as in Fig. 5].

Figure 11

Correlation indicators computed as a function of the size of the probing zone $R_{\text{free}}$. Top: Correlation with the first plastic rearrangement locations. Middle: decorrelation strain. Bottom: Average correlation.

Figure 12

Top row: Local yield stress maps of a GQ glass computed for different inclusion sizes $R_{\text{free}}$: (a) 5, (b) 7.5, (c) 10, and (d) 15. Bottom row: Corresponding local yield stress maps deduced from local minima obtained from the map computed for $R_{\text{free}}=5$ shown in panel (a).

Figure 13

Probability distribution function of the local yield stress for the three different quench protocols as a function of the inclusion size $R_{\text{free}}$: (a) HTL, (b) ESL, and (c) GQ. The lines correspond to the zoom-out process exemplified in the bottom row of Fig. 12, where the local yield stresses are deduced from maps computed with $R_{\text{free}}=5$.