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

The strength of FCC metals at low strain rates shows weak dependence on strain rate and temperature, increasing only about 10% with an order of magnitude increase in rate. However, a dramatic change occurs around 10⁴ s⁻¹, where strength increases 100 times more rapidly. The physical mechanism driving this transition, observed since 1984, remains unclear. Past studies have been hampered by the inability to separate instantaneous strength contributions (viscous effects, phonon drag) from structural contributions (changes in dislocation structure). This paper aims to clarify the nature of this transition and provide a unifying framework for plasticity across a wide range of strain rates. At low rates, plastic deformation occurs in discrete bursts, with dislocation dynamics exhibiting self-organized criticality (SOC). These intermittent flows arise from collective dislocation slip in avalanches, ranging in scale from single dislocations to 3D collective events. The independence of SOC from external tuning leads to the weak rate and temperature dependence of work hardening at low rates. However, high strain rates are likely to disrupt SOC dynamics. This paper investigates this disruption, hypothesizing that the work hardening driven strength transition is caused by the time limitation of self-organization—an idea supported by previous suggestions that avalanche disruption begins above 10⁴ s⁻¹. Existing Orowan-Taylor models, which assume independently slipping dislocations, don't predict such a transition. The authors aim to test this time-limited SOC hypothesis with novel high-rate experimental methods.

The paper reviews existing literature on the strain rate dependence of FCC metal strength, highlighting the abrupt transition around 10⁴ s⁻¹ and the ongoing debate about its underlying mechanism. It discusses the limitations of previous experimental approaches, such as the difficulty in distinguishing instantaneous and structural contributions to strength. The literature on avalanche plasticity at low strain rates and its connection to self-organized criticality is also reviewed, emphasizing the role of dislocation avalanches in influencing material properties, including work hardening and failure mechanisms like adiabatic shear banding. The authors note that avalanche dynamics are generally ignored in high-rate plasticity theories, with few exceptions like Hall-Petch effect descriptions. The authors mention previous work suggesting that dislocation avalanches and SOC are disrupted at strain rates above 10⁴ s⁻¹, making time-limited SOC a strong candidate for explaining the observed strength transition. The authors contrast this with the Orowan-Taylor model which does not predict a change in response at these rates and also review 2d simulations showing slip abandoning SOC above 10³ s⁻¹. These simulations highlighted increases in the metal's mechanical threshold.

To test plasticity models at high rates, the authors developed new experimental methods capable of distinguishing instantaneous and structural strength contributions in uniaxial compression experiments, from quasi-static rates up to 10⁵ s⁻¹. Below 10⁴ s⁻¹, standard split Hopkinson pressure bar (SHPB) and universal testing machine methods were used. Above 10⁴ s⁻¹, a direct impact Hopkinson pressure bar arrangement was employed to achieve rapid attainment of the target strain rate, overcoming limitations of conventional SHPB systems where wave dispersion during transit through the input bar delays specimen equilibrium. The authors empirically verified specimen equilibrium at the target strain rate through elastic wave measurements using photon Doppler velocimetry (PDV). Deformation was interrupted at a particular strain using a tungsten-carbide strain-limiting ring. To separate instantaneous and structural contributions, each specimen was immediately reloaded to yield at a much lower rate, removing any instantaneous strengthening mechanisms. High-purity copper specimens were deformed at various strain rates (10⁻² s⁻¹ to 10⁵ s⁻¹) and a fixed strain (10.00 ± 0.05%). The strength was measured before and after interruption. Experiments at elevated temperatures were also conducted to investigate the temperature dependence of the mechanical threshold. The authors analyzed avalanche dynamics around the strength transition, considering the time between avalanches and the duration of individual avalanches. They used a model of avalanche plasticity where avalanches correspond to slip bands caused by the rapid planar expansion of 'blade-like' Eshelby inclusions bound by a surface of edge and screw dislocations. The model allows for derivation of 3D avalanche properties and material behaviors matching experimental observations.

The experiments showed that the strength increase above 10⁴ s⁻¹ is due to a permanent structural change, evident as a strain-rate-dependent increase in early-stage work hardening. The temperature dependence of the mechanical threshold at a fixed post-transition rate was inconsistent with phonon drag as a dominant mechanism. The authors found that above a critical strain rate, there is insufficient time for large dislocation avalanches to synchronize, leading to a progressive disruption of self-organized criticality. The largest avalanches fail first, resulting in the formation of progressively finer dislocation networks and an increased rate of work hardening. The model predicts a 2/3 power law relationship between the initial rate of work hardening and the normalized strain rate above the transition. Experimental data on work hardening, normalized to the shear modulus and strain rate, strongly supports this 2/3 power law, collapsing onto a master curve across multiple temperatures. The model also accurately reproduces reload stresses across different temperatures, rates and strains, using only room temperature data and a saturation stress model. This accurate reproduction of reload stresses points to a separation between work hardening (system-scale, SOC) and dynamic recovery (atomic-scale) processes. The absence of significant change in saturation stress above 10⁴ s⁻¹ further supports this separation.

The findings demonstrate that the strength transition in FCC metals at high strain rates is primarily caused by the breakdown of large-scale self-organized criticality in dislocation dynamics due to time limitations. The model successfully accounts for the observed strain rate dependence of work hardening and the lack of significant changes in saturation stress above the transition. This work provides a unified description of plasticity across low and high strain rates, linking the collective behavior of dislocations to macroscopic material properties. The ability to predict high-rate behavior from low-rate data and microstructural observations is a significant advance. The model's success suggests that understanding the synchronization and activation of dislocation avalanches is critical for designing high-impact resistant materials, microscale forming, and ultrafine-grained metals.

The research presents a unified model for plasticity in FCC metals, integrating low-rate avalanche plasticity with high-rate dynamic plasticity. The model, based on the synchronization of dislocations into avalanches and their self-organization, accurately predicts material behavior at high strain rates using only low-rate information. The 2/3 power law relationship between the initial rate of work hardening and the normalized strain rate is a key finding, validated by the experimental data. Future research should focus on refining the model to predict the prefactor, by considering the effects of grain size, specimen size, and impurity content on avalanche confinement. Investigating the breakdown of self-organization in dislocation systems could enhance our understanding of SOC in various scientific fields.

The model focuses on high-purity copper and its applicability to other FCC metals or alloys needs further investigation. The assumption of an ellipsoidal avalanche geometry is a simplification; real avalanches may have more complex shapes. The model neglects other deformation mechanisms, such as twinning, which become important at higher strains, rates, or lower temperatures. The model parameters, while physically grounded, currently require experimental calibration.