Dislocation transport and intermittency in the plasticity of crystalline solids

When envisioned at the relevant length scale, plasticity of crystalline solids consists in the transport of dislocations through the lattice. In this paper, transport of dislocations is evidenced by experimental data gathered from high-resolution extensometry carried out on copper single crystals in tension. Spatiotemporal kinematic fields display spatial correlation through characteristic lines intermittently covered by plastic activity. Intermittency shows temporal correlation and power-law distribution of avalanche size. Interpretation of this phenomenon is proposed within the framework of a field dislocation theory attacking the combined problem of dislocation transport and long-range internal stress field development. Intermittency and transport properties show remarkable independence from sample size, aspect ratio, loading rate, and strain-rate sensitivity of the flow stress.

Figure 1

(Color online) (a) Cu single crystal oriented for multislip under uniaxial tension (gauge length: 30 mm, width: 5.5 mm, thickness: 5.5 mm, Schmid’s factor: 0.3, temperature: $20°\text{C}$, driving strain rate: ${\dot{ϵ}}_a=5\times {10}^{-4}{\text{s}}^{-1}$), macroscopic force vs time (main graph) and displacement vs time in six locations distant by 1 mm (inset). (b) Variations in axial strain rate about the driving strain rate ${\dot{ϵ}}_a$, as obtained from the lowest displacement curves in the stack in (a). Note that the size of the fluctuations can be larger than ${\dot{ϵ}}_a$. (c) Probability density (normalized to bin size) for event size in time series shown in (b). The dashed line indicates the power-law trend with slope $\tau =2$. (d) Intercorrelation between strain rates derived from displacement signals at adjacent locations $\left(x_i,x_{i+1}\right)$ and $\left(x_{i+1},x_{i+2};i=1,13\right)$ (blue dots); intercorrelation between strain rates derived from displacement signals at distant locations $\left(x_1,x_2\right)$ and $\left(x_j,x_{j+1};j=2,14\right)$ (red open circles). Same data except sample thickness: 2.3 mm and driving strain rate: $5\times {10}^{-3}{\text{s}}^{-1}$.

Figure 2

(Color online) Longitudinal fluctuations about the imposed strain rate in a space-time diagram during the elastoplastic transition. Dotted characteristic lines run from the left and right of the gauge length, reflecting intermittency and transport. The imposed strain rate is ${\dot{ϵ}}_a=5\times {10}^{-4}{\text{s}}^{-1}$. Fluctuations can be as high as $2.5\times {10}^{-3}{\text{s}}^{-1}$.

Figure 3

(Color online) (a) Stress vs time response during stage II linear hardening. The highlighted portion corresponds to the strain-rate plots shown in (b). (b) Successive frames of the strain-rate spatiotemporal field along the sample showing intermittent plasticity through dislocation transport.

Figure 4

Probability density of event size. The event size is defined either as a stress drop (filled circles; the dotted trend line shows a ${\tau }_s=1.2$ slope) or the maximum amplitude of stress rate bursts in the linear hardening region (open circles; the dashed trend line shows a $\tau =1.9$ slope).

Figure 5

(Color online) Model predictions for axial $\left(x_1\right)$ strain-rate fluctuations in a space-time diagram. The sample is a $13\times 13{\text{mm}}^2$ square in a glide plane subjected to equal shear rates $5\times {10}^{-4}{\text{s}}^{-1}$ on both sides. The figure shows the evolution in time of the strain-rate profile seen along the $x_2$ direction.

Figure 6

(Color online) (a) Polar dislocation density distribution showing self-organized loops in a $100\times 100\mu {\text{m}}^2$ square in the glide plane at 0.5% plastic strain. The driving strain rate is $5\times {10}^{-4}{\text{s}}^{-1}$. Note that the maximum fluctuation size is larger than the driving strain rate. (b) Time series ($50000$ data points) for the net shear strain rate (i.e., fluctuations about the applied strain rate) at a given point in (a). The closeup in the inset suggests self-similar structure. (c) Probability density of event size at three distinct points in (a). The event size is defined as the maximum strain-rate value during the event. The dotted line shows a $\tau =2$ slope.