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Four groups (1 × 10–5/s, 1 × 10–4/s, 1 × 10–3/s and 1 × 10–2/s) of triaxial compression tests at a confining pressure of 30 MPa are carried out to clarify the strain rate effect and damage evolution of siltstone. The elastic modulus, peak stress, strain energy evolution and failure modes of siltstone at different strain rates are analyzed, followed by an interpretation based on statistical damage theory. In the range of the quasi-static strain rate, as the strain rate increases, the elastic modulus, peak stress, total strain energy density and elastic strain energy density at the peak stress of siltstone increase, and the failure mode of the siltstone gradually change from single-plane shear failure to extensile failure. The plastic shear strain is adopted as the random variable of the Weibull distribution to formulate a new statistical damage model, which can better reproduce the post-peak characteristics of the stress–strain curve and interpret the damage evolution process of siltstone; this also reveals that the nonlinear evolution of random variables with axial strain should be considered to capture the post-peak stress–strain curve. The damage variable evolution curves are similar to the dissipated strain energy density curves, which are ‘L-shaped’. Finally, a simplified method for determining the influence of strain rate on damage at peak stress Dcr is proposed and discussed.HighlightsIn the range of the quasi-static strain rate, four groups of triaxial compression tests are carried out to reveal the strain rate effect of siltstone.Incorporating the plastic shear strain, a statistical damage model is proposed to better interpret the mechanical behaviour of siltstone.Based on the damage theory, a simplified method for determining the influence of strain rate on damage at peak stress Dcr is proposed.

To systematically study the influence of axial- and lateral-strain-controlled loadings on the strength and post-peak deformation behaviors of brittle rocks, four types of rocks (marble, sandstone, granite, and basalt) are tested under uniaxial and triaxial compressions, using a brittle hard rock testing system named Stiffman with high loading system stiffness. The test results show that the post-peak stress–strain curves of rock specimens under axial-strain-controlled loading are Class I, while those under lateral-strain-controlled loading are mostly Class II when the confining pressure is low. As confining pressure increases, the stress–strain curves change to Class I. Compared with that under lateral-strain-controlled loading, the failure of rock under axial-strain-controlled loading is more intense, the peak strength is higher, and the residual strength is lower. It is demonstrated that Class II post-peak stress–strain curves obtained by lateral-strain-controlled loading are caused by the unloading of the actuator in response to the servo-control system to keep the lateral strain rate constant. In the post-peak deformation stage, large dilation occurs, which leads to a sudden increase of the lateral strain rate; in order to keep the lateral strain rate at the set value under lateral-strain-controlled loading, the servo-control system must force the actuator to unload. When rock dilation occurs in the pre-peak deformation stage, unloading can occur before the peak strength, resulting in a decrease of peak strength compared with the peak strength obtained by axial-strain-controlled loading. It is found that the more brittle and dilatant a rock, the earlier the unloading of the actuator is, and the larger the peak strength decreases and the more obvious the Class II curve is.

Studies have shown that the deformation of Ti alloys is due to the competition between hardening and softening effects under dynamic loading. However, there are limited indicators of this behaviour throughout the complete stress–strain process. This study aims to quantify the impact of strain rate and temperature on the hardening/softening behaviour of Ti-6Al-4V using a split-Hopkinson pressure bar system over a range of 2000 to 7000 s −1 strain rates and temperatures from 25 to 800°C. Firstly, this study proposes an evaluation index of material hardening/softening behaviour based on the complete stress–strain curve and energy evolution characteristic. Further, the dynamic mechanical properties of Ti-6Al-4V are investigated through the analysis of the stress–strain relationship and fracture morphology. Finally, the hardening/softening index is calculated and analysed. The findings revealed that the fracture surface of the impact specimen displayed dimple-like and smooth features, that are significantly influenced by both temperature and strain rate. The stress–strain curves demonstrated that Ti-6Al-4V exhibits remarkable strain-rate strengthening, plastic increasing, and strain work hardening behaviour. The hardening/softening index B r decreases with an increase in strain rate. For specific strain rates of 3000, 5000 and 7000 s −1 , B r increases as the loading temperature rises from 25 to 400°C, but decreases when the loading temperature is increased to 600°C. At a strain rate of 2000 s −1 , B r increases monotonically until the loading temperature reaches ∼ 800°C. These observations are found to be related to the microstructural evolution at varying temperatures and strain rates. Graphic Abstract

This paper investigates the cracking and stress–strain behavior, especially the local strain concentration near the flaw tips, of rock-like material containing two flaws. A series of uniaxial compression tests were carried out on rock-like specimens containing two flaws, with strain gauges mounted near the flaw tips to measure the local strain concentration under the uniaxial compressive loading. Four different types of cracks (wing cracks, anti-wing cracks, coplanar shear cracks and oblique shear cracks) and seven patterns of crack coalescences (T1 and T2; S1 and S2; and TS1, TS2 and TS3) are observed in the experiments. The type of crack coalescence is related to the geometry of the flaws. In general, the crack coalescence varies from the S-mode to the TS-mode and then to the T-mode with the increase of the rock bridge ligament angle. The stress–strain curves of the specimens containing two flaws are closely related to the crack development and coalescence process. The strain measurements indicate that the local tensile strain concentration below or above the pre-existing flaw tip causes wing or anti-wing cracks, while the local compressive strain concentration near the flaw tip is related to the shear crack. The measured local tensile strain shows a jump at the initiation of wing- and anti-wing cracks, reflecting the instant opening of the wing- and anti-wing crack propagating through the strain gauge. During the propagation of wing- and anti-wing cracks, the measured local tensile strain gradually increases with few jumps, implying that the opening deformation of wing- and anti-wing cracks occurs in a stable manner. The shear cracks initiate followed by a large and abrupt compressive strain jump and then quickly propagate in an unstable manner resulting in the failure of specimens.

Metals with nanocrystalline grains have ultrahigh strengths approaching two gigapascals. However, such extreme grain-boundary strengthening results in the loss of almost all tensile ductility, even when the metal has a face-centred-cubic structure—the most ductile of all crystal structures 1 – 3 . Here we demonstrate that nanocrystalline nickel–cobalt solid solutions, although still a face-centred-cubic single phase, show tensile strengths of about 2.3 gigapascals with a respectable ductility of about 16 per cent elongation to failure. This unusual combination of tensile strength and ductility is achieved by compositional undulation in a highly concentrated solid solution. The undulation renders the stacking fault energy and the lattice strains spatially varying over length scales in the range of one to ten nanometres, such that the motion of dislocations is thus significantly affected. The motion of dislocations becomes sluggish, promoting their interaction, interlocking and accumulation, despite the severely limited space inside the nanocrystalline grains. As a result, the flow stress is increased, and the dislocation storage is promoted at the same time, which increases the strain hardening and hence the ductility. Meanwhile, the segment detrapping along the dislocation line entails a small activation volume and hence an increased strain-rate sensitivity, which also stabilizes the tensile flow. As such, an undulating landscape resisting dislocation propagation provides a strengthening mechanism that preserves tensile ductility at high flow stresses. A nanocrystalline metallic alloy with ultrahigh tensile strength and good ductility is achieved by introducing compositional undulation in a highly concentrated solid solution.

The purpose of this review is to discuss the development and the state of the art in dynamic testing techniques and dynamic mechanical behaviour of rock materials. The review begins by briefly introducing the history of rock dynamics and explaining the significance of studying these issues. Loading techniques commonly used for both intermediate and high strain rate tests and measurement techniques for dynamic stress and deformation are critically assessed in Sects.  2 and 3 . In Sect.  4 , methods of dynamic testing and estimation to obtain stress–strain curves at high strain rate are summarized, followed by an in-depth description of various dynamic mechanical properties (e.g. uniaxial and triaxial compressive strength, tensile strength, shear strength and fracture toughness) and corresponding fracture behaviour. Some influencing rock structural features (i.e. microstructure, size and shape) and testing conditions (i.e. confining pressure, temperature and water saturation) are considered, ending with some popular semi-empirical rate-dependent equations for the enhancement of dynamic mechanical properties. Section  5 discusses physical mechanisms of strain rate effects. Section  6 describes phenomenological and mechanically based rate-dependent constitutive models established from the knowledge of the stress–strain behaviour and physical mechanisms. Section  7 presents dynamic fracture criteria for quasi-brittle materials. Finally, a brief summary and some aspects of prospective research are presented.

Calcareous sand is a special marine soil rich in calcium carbonate minerals, characterized by brittle particles. It is, therefore, widely used as a filling material in the construction of islands and reefs. In this study, a series of cyclic tri-axial tests were conducted on calcareous sand taken from a reef in the South China Sea under different confining pressures and cyclic stress ratio (CSR). Then, applying the shakedown theory, the cumulative deformation of calcareous sand under a long-term cyclic load of aircraft was evaluated. Results showed that with the increase in the effective confining pressure, the stress–strain curves of calcareous sand showed a change from the strain-softening to the strain-hardening state; the volumetric strain of calcareous sand showed a change from shear shrinkage and then shear expansion to continuous shear shrinkage. Calcareous sand showed three different response behaviors under cyclic load: plastic shakedown, plastic creep and incremental plastic failure. With the plastic strain rate as the defining index, this study determined the critical CSR of calcareous sand under different shakedown response statuses and found them to increase with the effective confining pressure. The empirical formula for critical stress was established based on the fitting analysis of critical CSR under different confining pressures, taking the confining pressure as the variable. At the early stage of the cyclic load, calcareous sand samples were under compression. When the resilient modulus grew rapidly and the number of loading cycles continued to increase, the particles of calcareous sand samples were crushed, causing the fine particles to fill the voids among coarse particles, further compacting the samples and increasing the resilient modulus of calcareous sand samples. Hardin’s breakage potential model was adopted to quantitatively describe the particle breakage of calcareous sand samples before and after tests. The results indicated that calcareous sand samples produced obvious particle breakage when the CSR was small. As the CSR increased, the extent of the breakage of the sample particles first increased and thereafter stabilized. This study provides a theoretical reference for the assessment of the dynamic stability of calcareous sand subgrade subjected to traffic loads.

Accurate evaluations of rock brittleness are very significant in the engineering geology and geotechnical engineering fields. Most previous studies have adopted the stress–strain relationship to propose a series of indices for rock brittleness estimations but have seldom considered rock damage. Rock damage can be viewed as an energy dissipation process during rock deformation, which is closely related to rock brittleness. In this study, a new brittleness index (BI23) was proposed by considering rock damage, and the rock damage was calculated by the energy-based method. Then, the newly proposed rock brittleness index was validated by analyzing the variations in rock brittleness under increasing confining pressures and temperatures. The results indicate that the rock brittleness estimated by BI23 shows a significant drop in the case of increasing confining pressures and temperatures. To demonstrate its performance and advantages, a comparative study between the BI23 index and some previous indices was conducted by analyzing the stress–strain curves (SSC) of four rock types (e.g., limestone, marlite, feldspar lithic sandstone, and feldspathic quartz sandstone). The comparative study shows that the BI23 is able to produce more stable and consistent rock brittleness even for the same rock type under different tests, which is considered to be a major improvement over previous indices. Finally, the brittleness value distribution patterns of BI23 for normal and extreme conditions are discussed. It is suggested that the scope of rock brittleness evaluations under normal conditions should be defined to be between 0.5 (ductile) and 1 (brittle) in practical applications.

BackgroundLateral ankle sprains are common in indoor sports. High shoe–surface friction is considered a risk factor for non-contact lateral ankle sprains. Spraino is a novel low-friction patch that can be attached to the outside of sports shoes to minimise friction at the lateral edge, which could mitigate the risk of such injury. We aimed to determine preliminary effectiveness (incidence rate and severity) and safety (harms) of Spraino to prevent lateral ankle sprains among indoor sport athletes.MethodsIn this exploratory, parallel-group, two-arm pilot randomised controlled trial, 510 subelite indoor sport athletes with a previous lateral ankle sprain were randomly allocated (1:1) to Spraino or ‘do-as-usual’. Allocation was concealed and the trial was outcome assessor blinded. Match and training exposure, number of injuries and associated time loss were captured weekly via text messages. Information on harms, fear-of-injury and ankle pain was also documented.Results480 participants completed the trial. They reported a total of 151 lateral ankle sprains, of which 96 were categorised as non-contact, and 50 as severe. All outcomes favoured Spraino with incidence rate ratios of 0.87 (95% CI 0.62 to 1.23) for all lateral ankle sprains; 0.64 (95% CI 0.42 to 0.98) for non-contact lateral ankle sprains; and 0.47 (95% CI 0.25 to 0.88) for severe lateral ankle sprains. Time loss per injury was also lower in the Spraino group (1.8 vs 2.8 weeks, p=0.014). Six participants reported minor harms because of Spraino.ConclusionCompared with usual care, athletes allocated to Spraino had a lower risk of lateral ankle sprains and less time loss, with only few reported minor harms.Trial registration number NCT03311490.

Designing and printing metamaterials with customizable architectures enables the realization of unprecedented mechanical behaviors that transcend those of their constituent materials. These behaviors are recorded in the form of response curves, with stress-strain curves describing their quasi-static footprint. However, existing inverse design approaches are yet matured to capture the full desired behaviors due to challenges stemmed from multiple design objectives, nonlinear behavior, and process-dependent manufacturing errors. Here, we report a rapid inverse design methodology, leveraging generative machine learning and desktop additive manufacturing, which enables the creation of nearly all possible uniaxial compressive stress‒strain curve cases while accounting for process-dependent errors from printing. Results show that mechanical behavior with full tailorability can be achieved with nearly 90% fidelity between target and experimentally measured results. Our approach represents a starting point to inverse design materials that meet prescribed yet complex behaviors and potentially bypasses iterative design-manufacturing cycles. Mechanical behavior of a material is captured by a measured stress-strain curve upon loading. Here, the authors report a rapid inverse design methodology via machine learning and 3D printing to create metamaterials with mechanical behavior that replicates a user-prescribed stress-strain curve.