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Brain deformation caused by a head impact leads to traumatic brain injury (TBI). The maximum principal strain (MPS) was used to measure the extent of brain deformation and predict injury, and the recent evidence has indicated that incorporating the maximum principal strain rate (MPSR) and the product of MPS and MPSR, denoted as MPS × SR, enhances the accuracy of TBI prediction. However, ambiguities have arisen about the calculation of MPSR. Two schemes have been utilized: one is to use the time derivative of MPS (MPSR1), and another is to use the first eigenvalue of the strain rate tensor (MPSR2). Both MPSR1 and MPSR2 have been applied in previous studies to predict TBI. To quantify the discrepancies between these two methodologies, we compared them across eight in-vivo and one in-silico head impact datasets and found that 95MPSR1 was slightly larger than 95MPSR2 and 95MPS × SR1 was 4.85 % larger than 95MPS × SR2 in average. Across every element in all head impacts, the average MPSR1 was 12.73 % smaller than MPSR2, and MPS × SR1 was 11.95 % smaller than MPS × SR2. Furthermore, logistic regression models were trained to predict TBI using MPSR (or MPS × SR), and no significant difference was observed in the predictability. The consequence of misuse of MPSR and MPS × SR thresholds (i.e. compare threshold of 95MPSR1 with value from 95MPSR2 to determine if the impact is injurious) was investigated, and the resulting false rates were found to be around 1 %. The evidence suggested that these two methodologies were not significantly different in detecting TBI.

Solid–solid phase transformations can affect energy transduction and change material properties (e.g., superelasticity in shape memory alloys and soft elasticity in liquid crystal elastomers). Traditionally, phase-transforming materials are based on atomic- or molecular-level thermodynamic and kinetic mechanisms. Here, we develop elasto-magnetic metamaterials that display phase transformation behaviors due to nonlinear interactions between internal elastic structures and embedded, macroscale magnetic domains. These phase transitions, similar to those in shape memory alloys and liquid crystal elastomers, have beneficial changes in strain state and mechanical properties that can drive actuations and manage overall energy transduction. The constitutive response of the elasto-magnetic metamaterial changes as the phase transitions occur, resulting in a nonmonotonic stress–strain relation that can be harnessed to enhance or mitigate energy storage and release under high–strain-rate events, such as impulsive recoil and impact. Using a Landau free energy–based predictive model, we develop a quantitative phase map that relates the geometry and magnetic interactions to the phase transformation. Our work demonstrates how controllable phase transitions in metamaterials offer performance capabilities in energy management and programmable material properties for high-rate applications.

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.

AA2050-T84 alloy is widely used in primary structures of modern transport aircraft. AA2050-T84 is established as a low-density aluminum alloy with improved Young’s modulus, less anisotropy, and temperature-dependent mechanical properties. During flights, loading rate and temperature variation in aircraft engine subsequent parts are commonly observed. The present work focuses on the effect of loading rate and temperature on tensile and fracture properties of the 50 mm thick (2-inch) AA2050-T84 alloy plate. Quasi-static strain rates of 0.01, 0.1, and 1 s−1 at −20 °C, 24 °C and 200 °C are considered. Tensile test results revealed the sensitivity of mechanical properties towards strain rate variations for considered temperatures. The key tensile properties, yield, and ultimate tensile stresses were positive strain rate dependent. However, Young’s modulus and elongation showed negative strain rate dependency. Experimental fracture toughness tests exhibited the lower Plane Strain Fracture Toughness (KIC) at −20 °C compared to 24 °C. Elastic numerical fracture analysis revealed that the crack driving and constraint parameters are positive strain rate dependent and maximum at −20 °C, if plotted and analyzed over the stress ratio. The current results concerning strain rates and temperatures will help in understanding the performance-related issues of AA2050-T84 alloy reported in aircraft applications.

Material embrittlement is often encountered in machining, heat treatment, hydrogen and low-temperature conditions among which machining is strain-rate related. More strain-rate evoked embrittlement is expected in material loading processes, such as in high-speed machining and projectile penetration. In order to understand the fundamental mechanisms of the strain-rate evoked material embrittlement, this study is concerned with the material responses to loading at high strain-rates. It then explores the strain-rate evoked material embrittlement and fragmentation during high strain-rate loading processes and evaluates various empirical and physical models from different researchers for the assessment of the material embrittlement. The study proposes strain-rate sensitivity for the characterization of material embrittlement and the concept of the pseudo embrittlement for material responses to very high strain-rates. A discussion section is arranged to explore the underlying mechanisms of the strain-rate evoked material embrittlement and fragmentation based on dislocation kinetics.

This paper reviews the flow behavior and mathematical modeling of various metals and alloys at a wide range of temperatures and strain rates. Furthermore, it discusses the effects of strain rate and temperature on flow behavior. Johnson–Cook is a strong phenomenological model that has been used extensively for predictions of the flow behaviors of metals and alloys. It has been implemented in finite element software packages to optimize strain, strain rate, and temperature as well as to simulate real behaviors in severe conditions. Thus, this work will discuss and critically review the well-proven Johnson–Cook and modified Johnson–Cook-based models. The latest model modifications, along with their strengths and limitations, are introduced and compared. The coupling effect between flow parameters is also presented and discussed. The various methods and techniques used for the determination of model constants are highlighted and discussed. Finally, future research directions for the mathematical modeling of flow behavior are provided.

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.

In this study, we utilize C‐band Sentinel‐1 radar images from 2015 to 2022, combined with interseismic horizontal GNSS velocities, to construct large‐scale, high‐resolution, 3‐D velocity and strain rate maps over a vast region of southern Tibet. We show the distribution of prevailing dilatational strain accumulation along the seven major rift zones. Using 2‐D elastic dislocations invoking a two‐fault model in a Bayesian framework, we quantified the decadal extension rates across the seven rift zones, and we suggest a total extension rate of 18.4 ± 1.7 mm/yr, consistent with geological and geodetic estimates. The resulting strain rate maps, combined with the earthquake catalog, help us identify areas with high earthquake potential. Our study enhances our understanding of the present‐day tectonics and kinematics in southern Tibet and provides important constraints for seismic hazard assessment in this region. Plain Language Summary In this research, we used satellite radar images from 2015 to 2022 and GNSS data to study the crustal deformation and strain distribution in southern Tibet, where the Earth's crust is actively stretching due to the collision of the Indian and Eurasian plates and the extrusion of crustal materials. By analyzing high‐resolution 3D velocities, we provided new high‐resolution surface strain maps over southern Tibet. We found that the widespread dilatational strain is mainly localized along seven major N‐S trending rift zones. Seven major rift zones are experiencing extension at a total rate of 18.4 ± 1.7 mm/yr. The strain rate maps, combined with historical earthquakes, helped us identify fault segments that are more likely to host earthquakes in the future. By mapping deformation and strain in greater detail, we provided valuable data that can improve our understanding of kinematics and earthquake risk assessments in geologically complex southern Tibet. Key Points We present InSAR‐based, high‐resolution maps of 3‐D velocities and strain rates in Southern Tibet There is prevailing dilatational strain along seven rift zones in Southern Tibet, with a total extension rate of 18.4 ± 1.7 mm/yr We show the distribution and spatial variations of extension rates for seven rift zones

The primary goal of this study was to investigate the monotonic tensile behavior of high-density polyethylene (HDPE) in its virgin, regrind, and laminated forms. HDPE is the most commonly used polymer in many industries. A variety of tensile tests were performed using plate-type specimens made of rectangular plaques. Several factors can affect the tensile behavior such as thickness, processing technique, temperature, and strain rate. Testing temperatures were chosen at −40, 23 (room temperature, RT), 53, and 82 °C to investigate temperature effect. Tensile properties, including elastic modulus, yield strength, and ultimate tensile strength, were obtained for all conditions. Tensile properties significantly reduced by increasing temperature while elastic modulus and ultimate tensile strength linearly increased at higher strain rates. A significant effect of thickness on tensile properties was observed for injection molding specimens at 23 °C, but no thickness effect was observed for compression molded specimens at either 23 or 82 °C. The aforementioned effects and discussion of their influence on tensile properties are presented in this paper. Polynomial relations for tensile properties, including elastic modulus, yield strength, and ultimate tensile strength, were developed as functions of temperature and strain rate. Such relations can be used to estimate tensile properties of HDPE as a function of temperature and/or strain rate for application in designing parts with this material.

Earthquakes in the dry lower continental crust are enigmatic, as they require very high deviatoric stresses. Field observations suggest that stress amplification in rigid blocks surrounded by ductile shear zones leads to seismic failure: the jostling block model. Here we quantify this model by systematically testing numerically how variations in geometry, material properties, and loading conditions impact magnitude and rate of stress amplification. We demonstrate that bulk strain rate is the dominant factor controlling stress amplification. High strain rates of 10−10–10−12 s−1 cause stresses on the 102–103–MPa level within 100–102 years, while lower strain rates are insufficient to generate the stresses required for lower‐crustal earthquakes. While geometries and material properties play a subordinate role in causing stress amplification, tests with varying loading conditions show that pure shear is more effective in generating high stress amplifications compared to simple shear in the case of the given geometry. Plain Language Summary The pace of deformation controls if earthquakes occur deep in the Earth's crust. The deep section of the crust below the continents, the lower continental crust, is often very strong but may contain weak zones that surround strong blocks. To break the rocks and generate earthquakes at great depth, very high stresses are required. Earthquakes can form in the strong blocks because they build up stress while the weak zones deform. In this study we calculate how stress build‐up depends on a variety of geometrical and material properties. Most of the tested properties have a small effect on how much and how fast stress builds up. The most important factor controlling stress build‐up is how rapidly the weak zones deform and we find that, if deformation is fast, stresses high enough to generate earthquakes build up within a few years to hundreds of years. Key Points Strain rate controls the magnitude and rate of stress amplification Stress amplification to seismic failure in a matter of years to hundreds‐of‐years Lower‐crustal earthquakes caused by stress amplification achieved by a variety of geometric configurations and material properties