Free Surface Velocity Research Articles

Hugoniot elastic limit of single-crystal tantalum at normal and elevated temperatures subjected to extreme strain rates

Large-scale nonequilibrium molecular dynamic simulations of impact-induced shock propagation in defect-free single-crystal Ta along the [001] direction is presented, paying particular attention to the formation of elastic-plastic two-wave structures. Plastic deformation and elastic limits of Ta single crystals for various crystal orientations were investigated in a recent paper [R. Ravelo et al., Phys. Rev. B 88, 134101 (2013)]. In the present paper, a comprehensive study on the Hugoniot elastic limit (HEL) and its dependence on associated physical parameters is reported. The observed attenuation of an elastic precursor in a stress-wave profile led to proposing wide-range scaling relations for HEL decay applicable for ultrahigh strain rates. The influence of a lateral dimension on the spatial profile of a stress wave and time profile of free surface velocity (FSV) in the distinct appearance of a spike-valley feature and its consequence on scaling relations is investigated. Further, it is shown that the commonly used method of calculating HEL from a FSV profile is inadequate for submicron-thick samples at high strain rates. The necessary modifications are incorporated that resulted in comparable HEL values by two complimentary approaches of stress and FSV profiles. Simulations carried out for increasing impact strength revealed a power law dependence of HEL with strain rate. A methodology based on extrapolation of stress decay law and strain rate scaling constructed by us is proposed here for prediction of HEL pertaining to larger length samples. Very good agreement between the predicted values and available experimental results makes the modeling robust and applicable for a wide range of strain rates and length scales. Finally, the study is extended for elevated sample temperatures up to melting. Interestingly, yield strength is shown to follow an increasing trend up to a certain temperature, beyond which it decreases. An empirical relation is proposed here for expressing temperature dependence of HEL that explains the observed yield strength anomaly. Results are justified through thermal analysis of shear stress and evolution of dislocation density.

Chemical reaction zone measurements in pressed trinitrotoluene (TNT) and comparison with triaminotrinitrobenzene (TATB)

Photonic Doppler velocimetry and digital high-speed shadowgraphy have been used to characterize the chemical reaction zone parameters of pressed trinitrotoluene (TNT) samples with an initial density of 1.568 g cm−3. Comparison of the nanosecond time-resolved particle velocity histories of the free surfaces of detonating charges in air and light vacuum and of the interfaces between TNT detonation products and lithium fluoride or polymethyl methacrylate windows allow one to bracket the von Neumann spike pressure between 24.8 and 28.8 GPa. Our velocity waveforms confirm the two-step reaction pathway already observed in TNT, triaminotrinitrobenzene (TATB), and nitromethane, with a first fast energy release over 80 ns followed by a slower release over 250 additional ns. We consider the end of the first release zone as the locus of the Chapman–Jouguet (CJ) state, and the CJ pressure thus lies between 17.6 and 17.7 GPa. The energy release is not completed in this Jouguet plane but only after about ≈280 ns when the carbon cluster formation process ends. This corresponds to both the end of the slow release part of the interface velocity profiles and to the moment at which the free surface velocity profiles reach their maximum. Our shadowgraphy images confirm that carbon formation occurs very rapidly after the detonation breakout, in good agreement with previous time-resolved small-angle x-ray scattering measurements. The comparison with similar results previously obtained on TATB allows one to further highlight their similarities, which mainly result from their excess carbon production at late times.

On Stokes wave solutions

Using a perturbation method with the aid of MATLAB®Symbolic Toolbox, a new set of Stokes wave solutions, up to the fifth order, are derived. The new solutions are expressed in terms of free surface profile, velocity potential and wave celerity (or frequency dispersion relation). The solutions in the deep water limit are also deduced and discussed. The new solutions are compared with the existing solutions up to the fifth order. Differences appear among the existing and the present solutions, starting at the third order. The causes for the differences are identified. The characteristic differences were highlighted in the figures for high order solutions of Stokes waves. Numerical examples are provided to quantify the differences between the new fifth-order solutions and Fenton’s, and Skjelbreia and Hendrickson’s solutions.

Runup of breaking solitary waves on rough uniform slopes

An experiment consisting of 122 tests was conducted for breaking solitary waves on rough and smooth slopes. The constant water depth and the incident wave height were varied to cover a wide range of solitary wave nonlinearity, i.e., from 0.039 to 0.458. Free surface elevations, velocities and maximum runup heights were measured for four slope surfaces of different roughness. A surf similarity parameter proposed previously for smooth slopes was used to analyze the runup data on rough slopes. A reduction factor was introduced to compare the runup heights of breaking solitary waves on rough and smooth slopes and to unify the runup formula for smooth and rough slopes. The fitted reduction factor for each of the rough surfaces was less than unity (smooth slope) and in the range of 0.48–0.88. Existing runup formulas are used to confirm the accuracy and applicability of runup height predictions. The measured free surface elevations and velocities were analyzed to quantify the roughness effect on solitary wave runup. Laboratory observations reveal that the slope roughness noticeably influences the hydrodynamics in the surf and swash zones.

Molecular dynamics study on the shock induced spallation of polyethylene

Macroscopic experimental results of the plate impact tests of polymers are generally interpreted using the free surface approximation and the acoustic approximation. However, their validity over a range of shock pressures has not been thoroughly investigated yet. We conducted molecular dynamics simulations of plate impact tests of polyethylene to obtain molecular-level insights on those two common approximations associated with the interpretation of shock pressure and spall strength. Our results revealed that the free surface approximation could slightly underpredict the shock pressure in the polymer. The spall strength computed from the free surface velocity history can be significantly smaller than the actual tensile stress in the region of spallation.

Shock response of pre-existing spall damage in copper

We perform molecular dynamics simulations to investigate second-shock-induced recompaction and the subsequent re-spallation process in Cu with pre-existing spall damage. Compared with the conventional spalling of pristine Cu free of damage, a clear disturbance is observed in the free-surface velocity under the second shock of a pre-spalled Cu sample. This is due to the complex interactions of shock waves during the collapse of existing voids. Full recompaction accompanied by local melting of residual damage is also observed in our second-shock simulations. Secondary spallation occurs after the recompaction process, and the effect of secondary loading conditions, including different values of the peak shock pressure and different flat-top widths of the shock wave, on second spallation is taken into account. The findings clearly show that the spall strength of pre-spalled Cu is lower than that of pristine Cu under the same loading conditions when the loading time is sufficiently long. This result is evidence of the locally disordered state of the recovered sample. Re-solidification accompanied by atomic rearrangement is observed in the disordered region after second-shock compression.

Modelling the Velocity Distribution around the Hydraulic Structure

Intense pressure and velocity oscillations, as well as extensive air entrainment, make the investigation of the hydraulic leap created in stilling basins very difficult. Its intricacy, coupled with the real-world need for stilling basins to dissipate energy, puts physical modelling front and center. Despite stilling basins' essential role in engineering, bibliographic research has often focused primarily on the more familiar case of the classical hydraulic leap. Thus, a physical model of a typical USBR II stilling basin was used to investigate the features of the hydraulic leap in this work. Under this framework, the model investigated the hydraulic jump's free surface profile and velocity distribution. A series of experiments were conducted to compare the efficacy of modern tools like the time-of-flight camera with that of more conventional ones like the Pitot tube. Notwithstanding some of the stated constraints, the findings demonstrated a good depiction of the free sample surface and the velocity distribution. In addition, the used instruments demonstrated the substantial impact the energy dissipation devices had on the flow parameters. In comparison to traditional hydraulic leaps, significant variations in hydraulic jump form and maximum velocity locations within recorded vertical profiles were discovered.

Dependence of spallstrength on temperature, grain size and strain rate in pure ductile metals

When a shockwave, which can be generated by high velocity impact or explosive detonation, reflects from the free surface of a metal, it usually creates tensile stress inside the metal. While the tensile stress is large enough, voids nucleation, growth and coalescence happen inside the metal, causing the metal to spall. As one of the main contents of the spallation damage research, the spallation strength, which is often characterized by features of the free surface velocity history measured in spallation experiments, represents the maximum tensile stress that the material can withstand, and is actually a complex interaction among several competing mechanisms. Optimizing the spallation strengths of metals is important for their applications in the aerospace, automotive, and defense industries, and can be achieved by using the advanced manufacturing strategies, if we can know better the meaning and present analytic model of the spallation strength of metal. A large number of experiments show that the spallation strength of ductile metal is strongly dependent on the tensile strain rate, grain size and temperature of material. Based on the analysis of early spallation evolution and influence of grain size and temperature on the material, a simple analytic model of spallation strength is presented in this paper, which takes into account the effects of strain rate, grain size and temperature in materials. The applicability of this model is verified by comparing the calculated results from the model with the experimental results of spall strength of typical ductile metals such as high purity aluminum, copper, and tantalum.

Inelastic behavior of tungsten carbide at high pressures

Tungsten carbide (WC) is commonly used in plate impact experiments for generating high pressures due to its combination of high impedance and high strength. In pressure-shear plate impact (PSPI) experiments, WC is used as anvil plates, which sandwich a thin sample of the material of interest. At pressures up to its HEL (∼5 GPa), an elastic response may be assumed for the anvil plate, enabling a straightforward analysis to extract the shear strength of the material. At higher pressures, exceeding the HEL, the inelastic behavior of WC must be taken into account to extract the shearing resistance of the material of interest. We report the results of pressure-shear plate impact experiments conducted in the pressure range of 9–37 GPa. Two modeling approaches are considered to account for the inelastic behavior of WC: (1) an elasto-plastic response, accounting for strain and strain-rate hardening, and (2) a brittle-like response, which incorporates pressure hardening and damage. The material models were calibrated using numerical simulations to reconstruct the measured free surface velocity components in the normal and transverse directions. Both models effectively simulate the measured velocity profiles. Above 20 GPa, the brittle model appears to better capture the experimental trends. The calibrated models for WC enable the reconstruction of the experimental data from the current work, as well as from results reported by others, for a broad range of pressures.

Spall and recompression processes with double shock loading of polycrystalline copper

Spall fracture induced by interaction of multiple shockwaves in shock-loaded material is a key issue for both basic science and practical application. Here, shock induced spall and subsequent recompression process of polycrystalline copper are investigated using molecular dynamics simulations. Two significant velocity peaks are observed from the velocity history of free surface, which is explained through the complex propagation of waves and dynamic damage processes. The overshoot feature is also noted in the free-surface velocities on account of the shock response of the damaged region. Recrystallization occurs after secondary shock passes through the spall region. Meanwhile, the differences in recrystallization processes are observed for the different shock response states of the spall region induced by secondary shock waves. What's more, the reduction of grain sizes in the spall region is also revealed in our simulation, which is consistent with experiments.

A photogrammetry-based variational optimization method for river surface velocity measurement

The ease of access to media resources and computational power has recently generated interest in using vision-based approaches for hydraulic monitoring. A key challenge for non-intrusive, image-based hydrology measurement methods is incorporating different hydraulic variables as prior knowledge with image information. We propose a photogrammetry-based method called L1-Diffusion derived from the convection–diffusion equation commonly used in hydrodynamics with an additional regularization term to estimate the fluid motion field in the image plane, from which the free surface velocity can be further obtained using the photogrammetric projection relationship between the image plane and world coordinates. The inverse problem is used to discuss the relationship between the widely used space–time image velocimetry (STIV) and the proposed L1-Diffusion. To validate the proposed method, unmanned aerial vehicle (UAV) images as well as in-situ acoustic Doppler current profiler (ADCP) experiments were carried out. Based on comparison results with the ADCP measurement and vision-based flow field estimation, the newly proposed L1-Diffusion algorithm can accurately and efficiently estimate the free surface velocity of a river from the image sequences in a variety of scenarios.

Artificial stress method resistance to tensile instability in simulating metal microspalling

<p indent=0mm>Metal microspalling is a phenomenon caused by the tensile failure of near-surface materials in molten metals. To study the dynamics of metal microspalling, the simulation results should be quantitatively compared with the experimental results. However, because of the problem’s complexity, the quantitative simulation is challenging. In this paper, the finite element method is used to simulate the detonation-driven metal tin microspalling phenomenon. Because of some nonphysical phenomena in the simulation, it is challenging to compare the velocity of the free surface and the density distribution of the microspalling region to the displacement pin system (DPS) velocity and X-ray results in the experiment. Nonphysical phenomena, such as density and velocity discontinuity in simulating the microspalling region, can be reduced or eliminated via an artificial stress method that can resist tensile instability in simulation. An artificial stress limiter is used to limit its action range to the metal tensile region. The results show that the nonphysical phenomena in the tensile region of microspalling can be effectively restrained using the artificial stress method. The density and velocity continuity of the microspalling region are considerably improved, and the mesh separation of the free surface is effectively suppressed. The simulation results in the density distribution and free surface velocity correlate well with the tin microspalling experiment driven by detonation. The analysis shows that although the artificial stress method’s resistance to tensile instability is false stress that does not physically exist, it can achieve stable calculation, weaken or eliminate nonphysical phenomena. Moreover, it has the characteristics of strong action range limitation, small amplitude, and does not affect the original conservation. The work in this paper breaks through the bottleneck of quantitative comparison between simulation and experiment of metal microspalling; in addition, it provides an important simulation technique support for dynamic analysis and physical model research of metal microspalling.

Investigation of structure and mechanical properties under quasi-static and planar impact loading of aluminum composite reinforced with Al2O3 nanoparticles of different shape

We report on an investigation of the structure and mechanical properties under quasi-static and planar impact loading of Al-based nanocomposites reinforced with alumina nanoparticles of ball, plate and sheet shape fabricated through accumulative roll bonding (ARB) for 4 and 10 cycles. The distribution of the nanoparticles and structural characteristics of the matrix were revealed using transmission electron microscopy. The microhardness, ultimate and yield strengths and ductility under tension at the strain rate of 1 × 10−3 s−1 were measured as mechanical characteristics under the quasi-static loading. The maximum pressure of the impact compression, Hugoniot elastic limit and spall strength were calculated using the free surface velocity data recorded during the impact by aluminum flyer-plates with the impact velocity of 630 ± 30 m/s. Our results show that nanoparticles tend to agglomeration inside the composite. The size of agglomerates depends on the nanoparticle shape and a number of ARB cycles. Only the part of the ball-shaped nanoparticles can be distributed uniformly as separate nanoparticles after 10 ARB cycles. The introduction of the nanoparticles into aluminum assists in the structural refinement in the nanocomposites relative to alumina-free aluminum. This effect is mostly exhibited in the composite reinforced with the ball-shaped nanoparticles, some of which are distributed separately. The nanoparticle of different shape affects the mechanical properties ambivalently under both quasi-static and shock-wave conditions due to their various distribution in the matrix and different properties of the agglomerates. The spall strength decreases with the nanoparticle introduction because the particles and their agglomerates are the stress concentrators and crack origins.

Effects of Impeller Rotational Speed and Immersion Depth on Flow Pattern, Mixing and Interface Characteristics for Kanbara Reactors Using VOF-SMM Simulations

The Kanbara Reactor (KR) is a primary desulfurization technology in the hot metal pretreatment refining process that is widely employed in the modern steelmaking industry. The operating parameters of KR impeller immersion depth (IID) and rotation speed (IRS) have a crucial impact on the process performance and the desulfurization effect. Still, their influences have not been fully understood. This study systematically investigated the effects of IID and IRS on the flow pattern, mixing behavior, vortex core depth, and free surface characteristics for KR processes based on a 3D Volume of Fluid (VOF) model coupled with the sliding mesh method (SMM). The model was validated via scale-down water model experiments and then applied to the KR process, and simulations found that IID and IRS have different impacts on the flow pattern. Specifically, the discharge flow location moves downward with IID increasing, but the discharge strength and mean velocity hardly changes. Comparatively, the rise of IRS significantly increases the mean velocity, but few changes occur to the discharge flow position. Increasing IRS improves bath hydrodynamics, strengthens recirculation, and efficiently shortens mixing time, but IID has a neglectable effect on these features. The minimum mixing time is 55 s at a maximum IRS of 260 rpm. Moreover, the vortex core depth and free surface velocity visibly increase with the increase of IRS. Comparatively, IID has a limited effect on the flow and mixing behavior but directly impacts the distribution of recirculation regions at the axial direction and the velocity gradient on the free surface at the radial direction. Furthermore, the correlation equations of these critical parameters as a function of the operating parameters were obtained. The results from this study may provide references for operating optimizations and industrial practices of KRs.

Study on the length of diagnostic time window of CUP-VISAR

One-dimensional line velocity interferometer system for any reflectors (VISARs) acts as a significant diagnostic device in the process of the inertial confinement fusion, which can be used to measure the free surface velocity and the history of shockwave velocity in transparent media over time. But the lack of spatial information makes it impossible to measure the flatness of wavefront. The compressed ultrafast photography (CUP) system developed in recent years can realize two-dimensional ultrafast photography. We have designed a new two-dimensional VISAR diagnosis system by combining the CUP with line-VISAR, called the CUP-VISAR, which will be of great significance in two-dimensional shockwave front measurement. In this paper, a data simulation method of CUP-VISAR is proposed to study the length of diagnostic time window of the device. A series of original VIASR images are firstly generated and then the compressed images are simulated after image encoding and compression. Finally, two-step iterative shrinkage thresholding algorithm is used to reconstruct the compressed images, which results in corresponding reconstructed images. In order to verify the restoration effect of the algorithm, we calculate the velocity error and image correlation coefficient value. Finally the relationship between the image reconstruction quality and the number of images is preliminarily obtained, which provides reference for the actual experiment.

Effects of shock-induced phase transition on spallation of a mild carbon steel

Effects of shock-induced phase transition (PT) on spallation of a mild carbon steel are investigated via plate impact experiments at impact velocities of 0.433–1.163 kms−1. The free surface velocity history is recorded to deduce spall strength and wave propagation/interaction within the sample. Postmortem samples are characterized with scanning electron microscopy and electron backscatter diffraction. Detailed wave analysis indicates that the PT affects wave propagation and interaction. Such wave propagation and interaction finally lead to the change of spall strength and spall plane position for different stress ranges. In the peak shock stress range of 13–17 GPa (slightly above the PT stress), the release fan of the PT wave accelerates the free surface but is not involved in the major spallation, giving rise to an overestimation of spall strength by the conventional acoustic method. However, at higher shock stress above PT (>17 GPa), the release fans of the PT wave interact with each other and induce secondary spallation, and the acoustic method is applicable in such cases; the major spall plane moves to the sample central region, and multiple secondary spall planes appear. In addition, the reverse PT upon release facilitates nucleation and growth of deformation twins which subsequently act as nucleation sits of voids during spallation. With increasing shock stress, the spall damage exhibits mode transitions from brittle damage (transgranular cleavage cracks), to mixed brittle and ductile damage, and to ductile damage (void nucleation and growth).

Shock-induced spallation in single-crystalline tantalum at elevated temperatures through molecular dynamics modeling

The effects of initial temperature on shock-induced spalling behavior and damage evolution of single-crystal Tantalum were investigated using molecular dynamics simulation. The wave profiles show that shock pressure and temperature increment rise as initial temperature increases, which can be explained utilizing the Rankine-Hugoniot relationship. It is found that strain rate and the initial temperature has a substantial effect on the spall strength. The spall strength will decrease with initial temperature increases, and the competition between the strain rate hardening and temperature softening effect on spall strength is discussed. The radial distribution function analysis reveal that the classical spallation occurs under shock velocity 1.5 km/s and micro-spalling state with material partially melted or melted happened in higher shock velocity. The simulations show that shock-induced spalling of Tantalum is characterized by void nucleation, growth, and coalescence. The void evolution characteristics in classical spallation and micro-spalling are discussed. Furthermore, it is found that the initial temperature has a dramatic effect on the void evolution. The total voids number increases as the initial temperature rises. The characteristics of the free surface velocity profile are also disscussed.

Laser-driven shocks to explore the effects of aging on the adhesion of silver sintered on copper substrate

The adhesion of thin silver joints sintered on metallic substrates is a major issue for their potential use in power electronics modules. Here, we explore how laser shock-induced spallation can provide a way to study this adhesion. Due to wave interactions inside the shock-loaded assembly, dynamic tensile stresses are induced near the copper-silver interface, resulting in partial or full removal of the silver layer. In order to investigate the effects of aging on the interface strength, experiments are performed on both as-sintered and aged samples. Time-resolved measurements of the free surface velocity provide information on wave dynamics and interface failure, but finite-element simulations show that inferring a quantitative value of the bonding strength from these data is still challenging. Nevertheless, post-recovery observations provide original insight into the debonding process, including some complex effects of aging, consistent with trends reported in the literature.

On two approaches to the third-order solution of surface gravity waves

A third-order approximate solution for surface gravity waves in the finite water depth is studied in the context of potential flow theory. This solution corresponds to the steady-state part of a multidirectional irregular wave field and provides explicit expressions for the surface elevation, free-surface velocity potential, and velocity potential. The amplitude dispersion relation is also provided. Two approaches are used to derive the third-order analytical solution, resulting in two types of approximate solutions: the perturbation solution and the Hamiltonian solution. The perturbation solution is obtained by a classical perturbation technique in which the time variable is expanded in multiscale to eliminate secular terms. The Hamiltonian solution is derived from the canonical transformation in the Hamiltonian theory of water waves. By comparing the two types of solutions, it is found that they are completely equivalent for the first- and second-order solutions and the nonlinear dispersion, but for the third-order part only the sum–sum terms are the same. Due to the canonical transformation that could completely separate the dynamic and bound harmonics, the Hamiltonian solutions break through the difficulty that the perturbation theory breaks down due to singularities in the transfer functions when the quartet resonance criterion is satisfied. Furthermore, it is also found that some time-averaged quantities based on the Hamiltonian solution, such as mean potential energy and mean kinetic energy, are equal to those in the initial state in which the sea surface is assumed to be a Gaussian random process. This is because there are associated conserved quantities in the Hamiltonian form. All of these show that the Hamiltonian solution is more reasonable and accurate to describe the third-order steady-state wave field. Finally, based on the Hamiltonian solution, some statistics are given such as the volume flux, skewness, excess kurtosis, and non-uniqueness of the induced mean flow and mean surface.

The simulation of 3D hypervelocity spallation using a hydrocode PAGOSA with FLIP+MPM

In this work, a hydrocode PAGOSA with FLIP+MPM is presented and exercised to investigate the fracture in ductile material. The merit of PAGOSA with FLIP+MPM to solve the advection problem is first illustrated by a solid piston periodically moving in a sealed tube with air. Furthermore, the ability of PAGOSA with FLIP+MPM to capture the fracture in material is shown by a simple stretching fracture in ductile material. In both of two benchmark problems, the PAGOSA results and analytical solutions are also presented for comparison. Then PAGOSA with FLIP+MPM is used to model complex spall in ductile material, which is a challenging problem in engineering applications. The convergences of PAGOSA with FLIP+MPM—based on both the mesh size and the marker density—are investigated by monitoring free surface velocity. To further show the ability of PAGOSA with FLIP+MPM to predict fracture in ductile material, the numerical results are compared with the experimental results and other published numerical results. Moreover, the effect of the spall parameter in PAGOSA with FLIP+MPM on the numerical simulation is also analyzed by investigating the free surface velocity. Finally, the effects of the peak compressive stress, the tensile strain rate, and the loading rate on the spallation are further investigated. The numerical results show that PAGOSA with FLIP+MPM can improve PAGOSA's performance when applied to predicting fracture in ductile materials, and is robust enough to accurately predict spall fracture in ductile materials.