Yield Precursor Dislocation Avalanches in Small Crystals: The Irreversibility Transition

The transition from elastic to plastic deformation in crystalline metals shares history dependence and scale-invariant avalanche signature with other nonequilibrium systems under external loading such as colloidal suspensions. These other systems exhibit transitions with clear analogies to work hardening and yield stress, with many typically undergoing purely elastic behavior only after “training” through repeated cyclic loading; studies in these other systems show a power-law scaling of the hysteresis loop extent and of the training time as the peak load approaches a so-called reversible-to-irreversible transition (RIT). We discover here that deformation of small crystals shares these key characteristics: yielding and hysteresis in uniaxial compression experiments of single-crystalline Cu nano- and micropillars decay under repeated cyclic loading. The amplitude and decay time of the yield precursor avalanches diverge as the peak stress approaches failure stress for each pillar, with a power-law scaling virtually equivalent to RITs in other nonequilibrium systems.

Figure 1

Precursor avalanches present in the uniaxial quasistatic and unload-reload cyclic compression experiments on single crystalline Cu pillars. (a) Representative stress-strain data for displacement-controlled (DC) compression experiments on different diameter pillars. (b) A close-up of the first fast-avalanche induced unloading-reloading process in the 300 nm diameter pillar DC compression test. The first reloading starts to deviate from the linear elastic response at a strain of $\sim 0.017$, while at a stress lower than the updated “yield stress,” defined as the previous maximum stress. (c) The in-parallel ($P$) stress (${\sigma }_r$) and in-series ($S$) strain (${ϵ}_r$) reconstruction of the reloading process marked in (b), where dots and solid lines represent raw and interpolated data separately. (d) The non-Hookean reconstruction results obtained from displacement-controlled tests on seven identically prepared pillars for each size of micropillars. (e) Sample stress-strain and (f) the reconstructed non-Hookean stress-strain data for two representative load-controlled (LC) unload-reload compression experiments on $3\mu \mathrm{m}$ and 500 nm diameter pillars. The area of the shaded region represents the precursor dissipation for $3\mu \mathrm{m}$ pillars.

Figure 2

Precursor avalanches trained over cyclic loading in micropillars. (a) Left: Estimated true stress-strain response from an LC training experiment on a $3\mu \mathrm{m}$ diameter Cu pillar. Unloading and reloading stress-strain curves are marked in blue and red, respectively. The maximum stress is increased in five steps; the fifth step reaches the failure stress ${\sigma }_c$. At each step, 100 unload-reload cycles are prescribed. Right: pre- and post-test scanning electron microscope (SEM) images of this sample with the arrow and dashed line denoting the crystallographic slip lines on parallel planes characteristic of dislocation avalanches and glide in the enlarged images on the side. (b) The drift-corrected stress vs strain (see Supplemental Material, Sec. S4, for details [27]) during the second, fifth, and eighth cycles from data shown in (a) loaded to a maximum of $\sim 340\mathrm{MPa}$. Shaded area represents the energy dissipated through precursor avalanches, which decreases over cyclic loading.

Figure 3

Training experimental results showing precursor dissipation activity at different maximum stresses. (a) The precursor dissipation energy $U$ at each representative maximum stress that shows its decay with the number of loading cycles, $n$. (b) Characteristic decay time $\tau$ versus the normalized maximum stress ${\sigma }_{\max }$ estimated for $3\mu \mathrm{m}$ diameter copper pillars. Inset shows that the estimated steady-state $U_{\infty }$ is close to zero below the critical stress ${\sigma }_c$ and abruptly increases to $\sim 2–4\mathrm{kPa}$ when ${\sigma }_{\max }$ reaches ${\sigma }_c$. The deviation of the final point in (b) and the corresponding points in (c) from the expected power-law divergence is probably due to our method for estimating the steady-state value (see the Supplemental Material, Sec. S7 [27]). (c) A direct comparison of dislocation RIT behavior gleaned from the Cu micropillar compression experiments with that reported for a colloidal particle system in a sheared suspension [8], which provides evidence for a divergence of necessary cycle time $\tau$ to reach a reversible state, close to the critical failure stress ${\sigma }_c$.