Elastic Precursor Research Articles

Impact response of pseudoelastic nitinol

The response of polycrystalline nitinol with solely austenite structure was studied in three series of planar impact tests characterized by loading of the nitinol samples of 0.5–10 mm thickness by 1 mm thick aluminum impactor accelerated up to velocities of about 387, 429, and 567 m/s. In all the tests, the velocities of the free surfaces of the samples were monitored by a laser velocity interferometer. It was found that in all three test series, the amplitude of elastic precursor wave, being initially greater than 4 GPa, rapidly decays with the propagation distance down to ∼2.5 GPa, below which the decay is hindered by atomic clusters of the nanometer size. Based on the part of the velocity histories indicating the shock-induced austenite–martensite transformation, the initial, of about 2.5 × 103 s−1, and the maximum, up to 1 × 105 s−1, rates of the transformation were determined. As well, the impact stress slightly greater than 4 GPa was determined as that required for the onset of the B2 → B19′ transformation under shock loading. The unloading parts of the same velocity histories allowed a rough estimate of the fraction of the shock-transformed martensite and the elucidation of the virtually complete reversibility of the transformation.

Material heterogeneity as the origin for quasi-elastic ramping and unloading

Plate impact experiments are widely used to study materials under high strain rates and pressures. However, discrepancies often arise when attempting to simulate the free surface velocity at the back of the target, even with modern and advanced material models. This work focuses on two key experimental features: the smooth rise in the elastic precursor wave and the smooth decay of the elastic release wave. We show, through mesoscopic simulations, that these features can be accurately reproduced when material strength heterogeneity is considered. To validate our model, we simulate polycrystalline metals—tantalum and copper—as well as a heterogeneous metallic composite, tungsten heavy alloy. Our results demonstrate that by incorporating mesoscopic strength variations, either due to grain orientation or a composite phase, the smoothed velocity profiles observed experimentally can be simulated while maintaining consistency with uniaxial stress compression tests.

Plastic deformation of [001]-oriented single-crystal iron under shock compression: Effects of void size

The thermomechanical behavior of materials is known to be sensitive to preexisting defects in their microstructure. In this paper, non-equilibrium molecular dynamics simulations have been used to investigate the effects of the microvoid size on the plastic deformation in single-crystal iron shock-compressed along the [001] crystallographic direction. The higher the microvoid radius, the faster the kinetics of dislocations. Thus, as the microvoid radius increases, the plastic activity evolves from a regime where the deformation is dominated by twin activities to a regime where both twin and dislocation activities play an essential role and then to a regime where the deformation is dominated by dislocation slip. Furthermore, in both defect-free and defective initial crystal states, the elastic precursor wave is observed to decay with propagation distance, resulting in a constitutive functional dependence of the yielding pressure, σE, on the plastic deformation rate, ε˙p. In the regime where both deformation twinning and dislocation slip play important roles, the constitutive behavior is consistent with the original Swegle–Grady model and is in overall agreement with experimental data and thermomechanical simulations.

Effect of Dislocation Density on the Dynamic Strength of Aluminium

The effects of cold work on dynamic flow strength is studied through measurements of dislocation density and elastic precursor attenuation in shocked aluminium. High-purity aluminium and Al 6082 alloy samples are subjected to up to three passes of rotary swaging, a severe plastic deformation (SPD) technique, in order to produce large variations in starting material conditions. Detailed material characterisation is performed using electron backscatter diffraction (EBSD) and transmission electron microscopy (TEM) to determine the initial dislocation density, texture, grain boundary density and misorientation distribution. Variations in dynamic strength are studied through the evolution of the elastic precursor amplitude with propagation distance in specimens with thicknesses between 0.2 and 1 mm, shock loaded to impact stresses of about 4.6 GPa. It is shown that dynamic strength decreases with increasing initial dislocation density in both materials, demonstrating rare, quantified measurements of a reversal in the effect of dislocation density on yield strength at strain rates around 104 s-1 and above.

Elastic precursor softening in proper ferroelastic materials: A molecular dynamics study

Precursor elastic effects are investigated in a displacive anharmonic spring model and shown to extend greatly into the paraelastic phase. Weak precursor effects can be detected near 2Ttr, where Ttr is the ferroelastic transition temperature. The precursor effects become strong at T<1.7Ttr. Two effects were identified in our two-dimensional model: the symmetry-breaking strain e3 (ɛxy) leads to softening of the elastic modulus C33, while the nonsymmetry-breaking strain e1+e2 (ɛxx+ɛyy) leads to hardening of C11. The strain e3 is proportional to the order parameter and scales as |e1+e2| ∼e32. The temperature evolutions of the elastic moduli are surprisingly well described by power laws and Vogel-Fulcher equations. The power-law exponents are ∼−0.5 for ΔC33 and ∼−1 for ΔC11, Δ(C11+C12) and Δ(C11−C12). The Vogel-Fulcher temperatures are very similar, while the Vogel-Fulcher energies differ between the excess elastic moduli. The origin of the precursor effect is the evolution of short-range order in the paraelastic phase which gives rise to a characteristic local nanostructure. In the case of the symmetry-breaking strain, this microstructure resembles dynamical twinning patterns corresponding to the ferroelastic nanostructure, which weakens the material. In the case of the nonsymmetry-breaking strain, we find density fluctuations which make the material harder. Published by the American Physical Society 2024

Direct imaging of shock wave splitting in diamond at Mbar pressure

Understanding the behavior of matter at extreme pressures of the order of a megabar (Mbar) is essential to gain insight into various physical phenomena at macroscales—the formation of planets, young stars, and the cores of super-Earths, and at microscales—damage to ceramic materials and high-pressure plastic transformation and phase transitions in solids. Under dynamic compression of solids up to Mbar pressures, even a solid with high strength exhibits plastic properties, causing the induced shock wave to split in two: an elastic precursor and a plastic shock wave. This phenomenon is described by theoretical models based on indirect measurements of material response. The advent of x-ray free-electron lasers (XFELs) has made it possible to use their ultrashort pulses for direct observations of the propagation of shock waves in solid materials by the method of phase-contrast radiography. However, there is still a lack of comprehensive data for verification of theoretical models of different solids. Here, we present the results of an experiment in which the evolution of the coupled elastic–plastic wave structure in diamond was directly observed and studied with submicrometer spatial resolution, using the unique capabilities of the x-ray free-electron laser (XFEL). The direct measurements allowed, for the first time, the fitting and validation of the 2D failure model for diamond in the range of several Mbar. Our experimental approach opens new possibilities for the direct verification and construction of equations of state of matter in the ultra-high-stress range, which are relevant to solving a variety of problems in high-energy-density physics.

Shock Wave–Induced Dynamic Mechanical Behavior of Calcium Silicate Aluminate Hydrate at the Molecular Scale

Calcium silicate aluminate hydrate (C-A-S-H) is the main hydration product of cement mixed with industrial wastes. The purpose of this study is to understand the dynamic mechanical behavior and structural transformations of molecular-scale C-A-S-H induced by shock waves. Three C-A-S-H models with Al/Si ratios of 0.0, 0.1, and 0.2 are constructed and reactive molecular dynamics simulations are used to perform shock compressions with different shock velocities from 0.1 km/s to 3.6 km/s. The distributions of particle velocity, pressure, and density along the shock direction are calculated using the binning analysis method, allowing Hugoniot pressure-specific volume curves to be derived. The results reveal that shock waves may induce elastic, elastic-plastic, or shock Hugoniot responses in molecular-scale C-A-S-H, depending on the Al/Si ratio and the shock velocity. Below the Hugoniot elastic limit (HEL), higher Al/Si ratios cause the elastic wave to propagate farther due to the cross-linking effect of aluminate units. Above the HEL, higher Al/Si ratios give rise to a distinct two-wave structure characteristic comprising a plastic front and an elastic precursor. This characteristic becomes less pronounced as the shock velocity increases. Analysis of the molecular structural transformations of C-A-S-H revealed that the main atomic deformation behavior below the HEL involves a reduction of interatomic distances; above the HEL the main response is a densification of water molecules followed by a general collapse of the layered structure as the shock velocity increases.

Stress wave propagation and force transmission in polymeric closed cell foams subjected to air shock loading

Polymeric foams are widely used in sandwich structures to withstand blast loadings, serving as energy dissipators and force attenuators. This study explores the mechanics involved when an air shock interacts with closed-cell polymeric foam. The authors provide a comprehensive analysis of stress wave propagation and its evolution across the foam specimen by employing ultra-high-speed imaging and digital image correlation (DIC). Measurement of elastic, plastic, and shock wave velocities inside the impacted foam was conducted. A methodology to predict elastic precursor wave velocity is proposed, that considers the foam material’s behavior at higher strain rates. The predicted values were found to be in excellent agreement with the experimentally obtained values. Furthermore, the authors investigate the force transmission through foams as a function of bulk foam density under unconfined conditions. The results reveal that under shock loading intensities where foam specimens remained in their elastic regime, a transmitted force amplification was observed relative to the input load. However, when subjected to higher-intensity shock loading, substantial foam deformation occurs during the initial propagation of the shock wave. In these cases, the transmitted force measured was found to rely on the plastic and shock waves generated within the material. Additionally, as the loading persists, the force transmissibility ratio decreases beyond the initial stress wave propagation phase, thus demonstrating the material’s suitability for blast mitigation applications.

Revisiting the effect of shear stress on the γ→α phase transition of cerium under shock loading

The dynamic response of Cerium under high pressure and temperature is not clear due to the complexity of its phase diagram. Using large-scale atomistic simulations, we report the γ→α phase transition (PT) of Ce under shock loading. The PT behaviors predicted by present simulations are in reasonable agreement with previous experiments, thus confirm the validity of the interatomic potential. While the simulated wave velocities of elastic precursor are larger than experiments, the PT wave velocities agree well with the LASL data. In the T-P phase diagram, the Hugoniot states generally agree with the lower-pressure extrapolation of the higher-pressure experimental data, but differ significantly from each other orientations. By examining the ratio between the shear stress and hydrostatic stress, it is found that the anisotropic response of PT kinetics might be dependent on the variance of shear stress with lattice orientations. The present study suggests that the effect of shear stress usually ignored in shock wave experiments has to be treated seriously when studying the low-pressure γ→α PT of Ce.

Flowstress and Overstress Approaches to Dynamic Viscoplasticity

Viscoplasticity is mostly modelled by the flowstress approach, where the flowstress (Y) is a function of pressure, temperature, plastic strain and strain rate Y (P, T, ). For dynamic viscoelasticity the flowstress approach is used in hydrocodes together with the radial return algorithm, to determine deviatoric stress components in each computational cell and for each time step. The flowstress approach assumes that during plastic loading, the flowstress in stress space follows the current stress point (current Y). Unloading of a computational cell is therefore always elastic. The overstress approach to dynamic viscoplasticity was used in various versions in the 1950s and early 1960s, before the advent of hydrocodes. By the overstress approach a state point may move out of the quasistatic flow surface upon loading, and hence the term overstress. When this happens, the state point tends to fall back (or relax) onto the quasistatic flow surface through plastic flow, and the rate of this relaxation is an increasing function of the amount of overstress. In this paper we first outline in detail how these two approaches to dynamic viscoplasticity work, and then show an example for which the overstress approach has an advantage over the flowstress approach. The example has to do with elastic precursor decay in planar impact, and with the phenomenon of anomalous thermal strengthening, revealed recently in planar impact tests. The overstress approach has an advantage whenever plastic flow during unloading is of importance.

Role of temperature and preexisting dislocation network on the shock compression of copper crystals

Shock wave experiments show that many annealed face-centered cubic metals exhibit an anomalously increasing yield strength with temperature at high strain rates. Such an increase is usually associated with the multiplication of dislocations in the phonon friction mode, which slows with temperature. However, the effect of temperature on the structure of the shock wave and the yield strength of metal crystals, including those with microstructure, remains elusive. In this paper, we perform molecular dynamic simulations of shock-wave loading for [110] and [111] copper crystals of 0.45 and 0.80μm length in a wide range of temperatures between 100 and 1100K to understand the role of temperature and preexisting dislocations on the Hugoniot Elastic Limit (HEL). We show that, in ideal copper crystals, the elastic precursor exhibits a form of plateau, and the HEL almost does not change with shock propagation distance. However, at higher impacts, the perturbations of an elastic precursor are observed leading to fluctuations in the HEL value. We show that temperature dependencies of the HEL are strongly anisotropic. The HEL values tend to decrease with temperature for [110] perfect copper crystals, and to increase with temperature for [111] copper crystals. This surprising result is explained by the presence of dislocation substructures in a plastic wave in [110] crystals, which reduces the mobility of dislocations and makes the process of dislocation nucleation more dominant than in [111] crystals. Preexisting dislocations in copper crystals allow the HEL to decay much faster than in ideal crystals. In contrast to ideal [110] crystals, [110] crystals with dislocations exhibit increasing HEL values with temperature, as do [111] crystals. We find that the HEL dependence could be well approximated by a power law, but the decay power changes as the wave propagates. The decay power values lie between 0.5 to 0.7 in the second stage for both [110] and [111] crystals, which is consistent with experimental results with polycrystalline annealed copper. Our findings extend our understanding of temperature-dependent mechanical properties of materials under high strain rate and crystal plasticity.

Elastic precursor effects during Ba1−xSrxTiO3 ferroelastic phase transitions

Elastic softening in the paraelastic phases of ${\mathrm{Ba}}_{1\ensuremath{-}x}{\mathrm{Sr}}_{x}{\mathrm{TiO}}_{3}$ is largest near the transition temperatures and decreases on heating smoothly over extended temperature ranges. Softening extends to the highest measured temperature (850 K) for Ba-rich compounds. The temperature evolution of the excess compliance of the precursor softening follows a power law $\ensuremath{\delta}S\ensuremath{\propto}|T\ensuremath{-}{T}_{\mathrm{C}}{|}^{\ensuremath{-}\ensuremath{\kappa}}$ with a characteristic exponent $\ensuremath{\kappa}$ ranging between 1.5 in ${\mathrm{SrTiO}}_{3}$ and 0.2 in ${\mathrm{BaTiO}}_{3}$. The latter value is below the estimated lower bounds of displacive systems with three orthogonal soft phonon branches (0.5). An alternative Vogel-Fulcher analysis shows that the softening is described by extremely low Vogel-Fulcher energies ${E}_{a}$, which increase from ${\mathrm{SrTiO}}_{3}$ to ${\mathrm{BaTiO}}_{3}$ indicating a change from a displacive to a weakly order-disorder character of the elastic precursor. Mixed crystals of ${\mathrm{Ba}}_{x}{\mathrm{Sr}}_{1\ensuremath{-}x}{\mathrm{TiO}}_{3}$ possess intermediate behavior. The amplitude of the precursor elastic softening increases continuously from ${\mathrm{SrTiO}}_{3}$ to ${\mathrm{BaTiO}}_{3}$. Using power-law fittings reveals that the elastic softening is still $33%$ of the unsoftened Young's modulus at temperatures as high as 750 K in ${\mathrm{BaTiO}}_{3}$ with $\ensuremath{\kappa}\ensuremath{\simeq}$ 0.2. This proves that the high-temperature elastic properties of these materials are drastically affected by precursor softening.

Understanding the So Called “Anomalous Thermal Strengthening” of Elastic Precursor Decay Tests

Some time ago a Russian shock physics group performed planar impact tests on thin metal targets (1mm or less) at different temperatures. They were able to observe the elastic precursor wave amplitudes at different target thicknesses and for different target temperatures. They concluded from the test results that the elastic precursor wave decay rate was lower for higher test temperatures, or as they phrased it, dynamic strength increases with temperature. Such a behavior is counter intuitive, as we are used to believe from our experience with quasistatic viscoplastic response, that strength should decrease with temperature. Accordingly, these results of the Russian group were referred to by them as "Anomalous Thermal Strengthening". Our purpose here is to reproduce such test results using a hydrocode. To this end we do the following: 1) we change the way our hydrocode computes dynamic viscoplastic response, from the usually used radial return approach to the more appropriate overstress approach; 2) we compute a planar impact example and reproduce an elastic precursor decay behavior; 3) we show how the quasistatic part of the response can be separated from dynamic part; and 4) we make the quasistatic and the dynamic response parts depend separately on temperature, and in this way we’re able to reproduce the elastic precursor wave increase with temperature, as observed in the above mentioned tests. &nbsp

Влияние температуры на динамический предел упругости и откольную прочность свинцово-висмутового сплава при давлении ударного сжатия до 2.4 GPa

Measurements of the Hugoniot elastic limit and the spall strength of the eutectic alloy Bi – 56.5 mass%, Pb – mass43.5% were carried out at sample temperatures in the range of 20 0C – 109 0C based on registration and analysis of the evolution of shock compression pulses of various amplitudes. It is shown that an increase in the temperature of the samples leads to a decrease in the Hugoniot elastic limit by 25%, and the spall strength of the alloy under study – by 30%, regardless of the strain rate. An increase in the strain rate by two orders of magnitude leads to an increase in the spall strength by about three times. Approximation power-law dependences of the decay of the elastic precursor on the thickness of the samples and the spall strength on the strain rate before fracture at normal and elevated temperatures are constructed.

The effect of temperature on the Hugoniot elastic limit and the spall strength of a lead-bismuth alloy at the pressure of shock compression up to 2.4 GPa

Measurements of the Hugoniot elastic limit and the spall strength of the eutectic alloy Bi --- 56.5 mass%, Pb --- 43.5 mass% were carried out at sample temperatures in the range of 20-109oC based on registration and analysis of the evolution of shock compression pulses of various amplitudes. It is shown that an increase in the temperature of the samples leads to a decrease in the Hugoniot elastic limit by 25%, and the spall strength of the alloy under study --- by 30%, regardless of the strain rate. An increase in the strain rate by two orders of magnitude leads to an increase in the spall strength by about three times. Approximation power-law dependences of the decay of the elastic precursor on the thickness of the samples and the spall strength on the strain rate before fracture at normal and elevated temperatures are constructed. Keywords: lead-bismuth eutectic alloy, shock waves, deformation, temperature, Hugoniot elastic limit, spall strength.

A comparative investigation of shock response in high entropy Cantor alloys by MEAM and LJ type potentials

We assess the effect of interatomic potential in determining the shock response of the high entropy Cantor alloy through large scale non-equilibrium molecular dynamics simulations. In literature, two different potentials have been suggested for modelling Cantor alloys at atomic scale – the 2NN MEAM potential, henceforth termed as the MEAM potential, and the Lennard-Jones (LJ) type-potential. We show that the MEAM potential results in a different shock response than the LJ type-potential – (i) the shock wave speed and the pressure behind the shock front are smaller for MEAM than the LJ type-potential, (ii) no twinning is observed with the MEAM potential along the [100] direction unlike that with the LJ type-potential, and (iii) the shock induced phase transition from FCC to HCP is observed along the [100] direction only with the LJ type-potential. Our numerical results with the LJ type-potential are in agreement with the experiments as well, where twinning and phase transition have been observed. Looking at the morphology of the post-shocked samples at different times, we identify the mechanism behind the phase transition. Lastly, we subject the post-shocked structure to uniaxial tensile tests to show that for the [111] direction, a greater fracture strain than the unshocked structure is obtained, suggesting a higher fracture toughness. • Used 2-NN MEAM and LJ-type potential to study the effect of shock loading in Cantor alloy. • The LJ-type potential results in higher shock pressure and shock speed as compared to MEAM. • The elastic precursor wave is overdriven by the plastic wave along [100], unlike the [110] and [111] directions. • Plastic deformation by slipping and twinning in [100] direction is observed with LJ-type, consistent with experiments, unlike the MEAM potential. • Disordered phases along [110] and [111] directions increases the fracture toughness in after-shocked samples.

Shock compression of magnesium alloy by ultrashort loads driven by sub-picosecond laser pulses

The shock compression of magnesium (Mg-4Al-2Zn) alloy polycrystalline films on glass under ultrashort loads driven by sub-picosecond laser pulses was investigated. The continuous diagnostics of motion and reflectivity changes of the free rear surface of the samples was carried out in the picosecond range (≤200 ps) in a single pulse mode using ultrafast spectral interferometry. We present the data on elastoplastic shock wave evolution at a propagation distance of several hundreds of nanometers, elastic precursor decay, shear, and tensile strengths at the extreme strain rate of ∼109 s−1.

Assessment of the time-dependent behavior of dislocation multiplication under shock loading

In this study, we proposed a method, which is a combination of qualitative analysis and quantitative analysis, to establish dislocation-based plasticity model, and assessed the time-dependent behavior of dislocation multiplication under shock loading. First, the dimensional analysis method is applied to generate the governing equation of dislocation multiplication rate, and ten potential governing equations are derived. Second, through the theoretical analysis of the mechanical response on the shock front, each time-dependent governing equation is directly integrated on the shock front, and the explicit expression of dislocation density with respect to the applied stress is derived, which is further quantitatively connected to the power scaling characteristics of the plastic front. It's found that only three of the ten equations, including two common equations and one newly derived equation via the dimensional analysis method, can predict the appropriate scaling coefficient of the plastic front. Third, three common equations and the newly derived equation are implemented into a dislocation-based constitutive model, and we compared results, including the velocity profiles and dislocation densities, calculated using the model associated with each equation and experimental results to evaluate the accuracy and efficiency of each equation. Through the comparison of the quantitative relation of the power scaling characteristics of the plastic front, and the elastic precursor decay, it's concluded that the newly derived stress-dependent governing equation is more suitable for shock-loading process than the commonly used equations by previous constitutive models.

Modeling the Fourth Power Law Response of Viscoplastic Materials

A standard way to parametrize a steady structured shock going through a viscoplastic material, is by plotting the stress jump across the wave (Ds) as function of its maximum strain rate () plotted on a logarithmic scale. Doing this we usually get a straight line, which means that . Grady and coworkers noticed in the 1980s that for many materials b=4. Since then, several workers in the field tried to understand this unusual behavior, but they were not able to arrive at any plausible explanation. Recently, Grady revisited his fourth power law with an extensive review, including an outline of various mechanisms proposed as possible explanations of this law. But still, none of these mechanisms is able to explain the fourth power law response. In view of this background, we use here the fourth power law to calibrate the strain rate component of the constitutive response of Al 6061-T6. To this end, we integrate the steady viscoplastic equations (developed long ago) describing the propagation of a plane wave from the elastic precursor level to the stress plateau level. We find that in order to reproduce the fourth power law, the plastic deformation rate has to depend on the deviator overstress by: , with α=2.38. But for high shock levels, when the elastic precursor is overdriven, using this value of α leads to a higher value of the slope b, close to b=6. This result has yet to be verified experimentally.

Dynamic response of YAG polycrystalline and single‐crystal transparent ceramics: Experiments and mesoscopic simulations

AbstractDue to superior mechanical and optical properties, yttrium–aluminum–garnet (YAG) is emerging as a potential transparent armor material. Its dynamic behavior, however, remains largely unexplored. In this work, both impact experiments and mesoscopic simulations have been performed to better understand the dynamic response of YAG polycrystalline and single‐crystal transparent ceramics. Experimental results demonstrate that the two samples have remarkably different dynamic behaviors with the increasing shock pressure, in which the Hugoniot elastic limit basically keeps unchanged (∼14 GPa) in YAG polycrystalline ceramic, whereas it varies from 11 to 31 GPa in YAG single crystals. Moreover, elastic precursor decay is visible only in single‐crystal samples. Mesoscopic simulations with a lattice‐spring model reveal that the difference arises from the distinct fracture mode in the two samples. Less damage and coarser fragmentations in YAG single crystal also suggest a probability of better ballistic resistance as transparent armor materials.