Time limited self-organised criticality in the high rate deformation of face centred cubic metals

Face-centred cubic (FCC) metals show very weak dependence of strength on strain rate and temperature at low rates, but exhibit an abrupt transition near 10^4 s^-1 above which strength increases much more rapidly. The physical origin of this transition has been debated, particularly whether it is due to instantaneous (e.g., viscous/phonon drag) effects or structural evolution of the dislocation network. At low rates, plasticity proceeds via intermittent dislocation avalanches consistent with self-organised criticality (SOC). High strain rates are expected to disrupt SOC because the imposed timescale can become comparable to avalanche formation and duration times. The study aims to determine whether the high-rate strength upturn is structural (mechanical threshold) in origin and to develop a physical framework that unifies low-rate avalanche plasticity with high-rate behaviour by incorporating time-limited self-organisation of dislocations.

Prior work established weak rate sensitivity in FCC metals at low rates and reported a strength upturn near 10^4 s^-1 (Follansbee et al.). SOC and dislocation avalanches explain scale-free intermittent plasticity, strain bursts, and work hardening in slowly driven conditions (Weiss; Dimiduk; Brown). High-rate plasticity theories often neglect avalanche dynamics, focusing on independently moving dislocations; such models do not predict a transition at ~10^4 s^-1. Acoustic emission suggests SOC disruption above ~10^4 s^-1 (Brown). Simulations indicate a move away from SOC at high rates (Song et al.). Microstructural studies at high rates report refined dislocation structures, consistent with altered collective slip. The Orowan-Taylor framework and phonon drag become important at yet higher rates or near melt. Mechanical threshold stress (MTS) theory provides a framework to separate structural and instantaneous contributions to flow stress (Follansbee & Kocks).

• Materials: High-purity oxygen-free copper (grade C103), annealed, grain size 16–20 µm. Cylindrical specimens with fixed aspect ratio; smallest 500 µm thick and 1.5 mm radius to minimise friction effects.
• Loading protocols: Uniaxial compression over strain rates from 10^-2 s^-1 to 10^5 s^-1. For rates ≥10^4 s^-1, a photon Doppler velocimetry (PDV)-instrumented direct impact Hopkinson pressure bar (DIHPB) was used to achieve rapid rise to target rate without shock. For lower rates, conventional SHPB and universal testing machine methods were used.
• Equilibrium verification: PDV measured elastic waves in striker and output bars to verify specimen equilibrium at target rates. Dynamic equilibrium achieved for >90% of deformation below 5×10^4 s^-1.
• Strain limitation and reload: Deformation interrupted at true strain 0.10±0.005 using a tungsten-carbide strain limiting ring. Immediately after high-rate loading, each specimen was reloaded quasi-statically at 10^-2 s^-1 to yield, isolating the permanent (structural) contribution by removing instantaneous rate/temperature effects.
• Thermal management: Strain limitation reduced adiabatic heating to ~10 K. Additional tests at elevated temperatures (up to ~300 °C) probed temperature dependence of the mechanical threshold.
• Data and analysis: Flow stress measured just before interrupt (high-rate), and yield upon quasi-static reload to estimate mechanical threshold stress (MTS). Additional interrupted tests over strains 0–1 and multiple temperatures enabled extraction of the initial work hardening rate Θ(ė,T), accounting for dynamic recovery and saturation (details in Supplementary Methods Section 4). Strain rates and stresses were normalised by shear modulus and sound speed to construct master curves.
• Modelling: A time-limited SOC framework was developed. Key elements include: (i) avalanches as 3D Eshelby-like ellipsoidal inclusions (aspect ~1:30) bounded by dislocations; (ii) probability of precursor size L following P(L)∝L^-2; (iii) eligible slip volume dominated by largest avalanches; (iv) synchronisation time for collective slip scaling as t_γ∝L^3/C_s, with an estimate t_γ≈√(2π/3)·(4π L_a^2)/(d b C_s); (v) largest in-plane axis L_a inferred ~100–300 nm; slip-normal axis L_c 3.1–10 nm; leading to predicted t_γ2–80 µs. Combining spatial/time limits gives Θ≈Θ_0·max[1,(ė/ė_c)^{2/3}], with temperature dependence embedded via sound speed and material parameters. Model parameters were fitted to room temperature data and then applied across temperatures.

• Structural origin of the high-rate strength upturn: The increased flow strength observed above ~10^4 s^-1 persisted upon quasi-static reload (10^-2 s^-1), indicating a permanent increase in mechanical threshold stress rather than an instantaneous viscous effect.
• Temperature dependence excludes phonon drag: At fixed post-transition rates, the mechanical threshold decreased with increasing temperature, inconsistent with phonon drag-controlled strengthening in the studied regime.
• Master curve and scaling: Normalising Θ by shear modulus and ė by sound speed collapses data from 10^-2–10^5 s^-1 and multiple temperatures onto a single curve. Above the transition, Θ scales with ė^{2/3}; best-fit slope 0.68±0.04 matches the predicted 2/3 power law.
• Transition rate and parameters: At 295 K, the transition occurs at ė_c=6900±300 s^-1. Extracted parameters: θ=1.68±0.03 GPa and λ=3.03±0.07 m. The full model (single fitted parameter from room temperature) reproduced reload stresses across all temperatures, rates, and strains.
• Synchronisation timescale: Using inferred avalanche dimensions and copper properties, the largest avalanche synchronisation time was estimated at 2–80 µs, consistent with acoustic emission intervals and shear band temperature-rise delays.
• Saturation stress unchanged: No significant change in saturation (plateau) stress above 10^4 s^-1, implying the rate-dependent increase in work hardening is decoupled from dynamic recovery mechanisms.
• Microstructural implications: Time-limiting eliminates the largest avalanches first, introducing an effective length scale, promoting finer dislocation networks, and increasing work hardening; consistent with observed replacement of cellular structures by more random dislocation distributions at higher rates and increased isolated dislocations within cells.
• Broader consistency: The framework explains observed activation volume reductions at high rates and grain-size-dependent shifts of the transition; predicts vanishing grain-size effects when the time-limited regime dominates at ultrahigh rates.

The study directly addresses whether the high-rate strength upturn in FCC metals originates from instantaneous drag or from structural evolution. By interrupting high-rate deformation and reloading quasi-statically, the persistence of strength increases demonstrates a structural (mechanical threshold) basis. The temperature trend—decreasing mechanical threshold with increasing temperature at fixed high rate—contradicts phonon drag dominance in this regime, further supporting a structural mechanism. The proposed time-limited SOC model posits that at high strain rates there is insufficient time for large avalanche precursors to synchronise, progressively removing the largest events that dominate slip-eligible volume and microstructural self-organisation. This predicts a higher initial work hardening rate with strain rate, following Θ∝ė^{2/3} beyond a transition ė_c. The experimental master curve across rates and temperatures matches this scaling and a single-parameter fit captures the full dataset. The unchanged saturation stress across the transition supports a separation between system-scale SOC-driven work hardening and atomistic-scale recovery. The model unifies low- and high-rate behaviour and rationalises observations across different microstructures and grain sizes via the interplay of spatial and temporal limits on avalanches.

The work establishes that the strength transition near 10^4 s^-1 in high-purity FCC copper is structural in origin, arising from time-limited self-organisation of dislocation avalanches. A physical model linking avalanche synchronisation time to the rate of work hardening predicts a Θ∝ė^{2/3} scaling above a temperature-dependent transition rate. Experiments using PDV-instrumented direct impact Hopkinson bars with strain-interrupt and quasi-static reload protocols validate the model: the strength upturn persists upon reload, a master curve collapses data across rates and temperatures, and fitted parameters reproduce stresses across conditions. The framework unifies SOC-based avalanche plasticity with high-rate dynamics and provides insight into microstructural evolution under fast loading. Future directions include predicting the model’s prefactor from microstructural constraints (grain size, specimen size, impurity population/type), direct characterisation of avalanche precursor dimensions and dynamics, extending the approach to other FCC materials and to regimes where alternative mechanisms (twinning, nucleation-limited slip, phonon drag) become dominant, and leveraging the model to forecast transitions between deformation modes.

• Material and state: Findings are for annealed, high-purity copper with 16–20 µm grains and limited to 10% strain; applicability to other alloys, prior work states, and grain sizes requires further validation.
• Experimental constraints: In-plane avalanche precursor axes cannot be directly measured; dimensions inferred from slip-band geometry and aspect ratios. Adiabatic heating was minimised but not eliminated (~10 K).
• Model approximations: The model neglects the minimum avalanche size in the simplified scaling, which can lead to a formal divergence in Θ when only single-dislocation slip remains; in reality, other mechanisms (twinning, nucleation-limited processes, drag) intervene. A key prefactor (A/λ) is fitted rather than predicted from first principles.
• Regime coverage: Conclusions pertain to strain rates up to ~10^5 s^-1 and temperatures up to ~300 °C. At higher rates (>10^6 s^-1) or near melting, phonon drag likely dominates and may further disrupt SOC, beyond the scope of the present measurements.