Shock-induced Plasticity Research Articles

Molecular dynamics simulation of shock-induced plastic deformation and spallation behavior of Cu/Cu64Zr36 crystalline/amorphous composites

Molecular dynamics (MD) simulations were performed to study the shock-induced plastic deformation and spallation failure of Cu/Cu64Zr36 crystalline/amorphous composites with a pre-existing void. The results show that the pre-existing void collapses always perpendicular to the direction of the shock loading, regardless of whether the shock direction starts from the crystalline phase or the amorphous phase. The Cu/Cu64Zr36 composites are more prone to spallation failure when the shock starts from the Cu crystalline phase. The changes of dislocation density and shear transformation zone (STZ) activation are closely related to the magnitude and direction of shock velocity. When the shock velocity reaches 2.0 km/s, dislocations in the crystalline phase disappear, dislocation density is close to zero and STZs activate throughout the entire amorphous phase. In addition, regardless of the shock velocity and direction, no shear bands are generated in the Cu/Cu64Zr36 composites under shock loading, which is significantly different from the case of tensile loading.

Orientation effects on shock-induced plastic deformation in FeNiCoCu high entropy alloy

FeNiCoCu high-entropy alloys (HEAs) demonstrate promising potential for widespread use in structural and functional applications. However, a thorough understanding of dynamic deformation processes in FeNiCoCu HEA is limited due to technological constraints in detecting real-time microstructural developments at the atomic level. This study examines the shock-induced plastic deformations in the equiatomic FeNiCoCu HEA, focusing on crystallographic orientation and particle velocity, using nonequilibrium molecular dynamics simulations. We obtained the P−V/V0, P−T, P−Up, and Us−Up Hugoniot relations and evaluated their anisotropy. The shock velocity, stress, and shear stress exhibit orientation dependence due to the differences in the plastic deformation mechanism. For shock loading along [100] orientations, dislocation dominates at lower shock intensities. However, a phase transition from face-centered-cubic (FCC) to body-centered-cubic becomes the primary plastic deformation at high shock intensity. For shock loading along [112¯] and [111] orientations, the generation of disordered structures and dislocation activities is revealed to play an important role in the development of localized plastic deformation. Moreover, the competition of disordered and hexagonal-close-packed (HCP) atoms is observed. The transition from FCC to disordered atoms provides nucleation sites for dislocations, and the slip of dislocations around disordered atoms leads to the formation of HCP structures. These findings are very helpful for learning the dynamic deformation behavior of FeNCoCu HEA.

Shaping sub-10 nm metallic nanogaps by laser shock-induced extreme plasticity

The nanoscale metallic gaps generate the unique local electromagnetic field enhancement effect due to the strong electromagnetic coupling between closely spaced nanostructures. Besides the conventional material deposition or removal strategy, laser shock reshaping (LSR), based on the ultrahigh-strain-rate deformation of the nanometals, provides a viable process for ultrafine-gap fabrication. However, the extreme deformation behavior of nanometals during the high-speed forming process, which significantly determines the forming precision and quality, is largely unexplored. In this study, the scalable fabrication of the nanogaps was achieved using the LSR process; the extreme deformation behaviors of the nanometals were comprehensively investigated. The results indicated that the geometry of the primary structure and the LSR process will signally affect the microstructure evolution and deformation behavior of the sidewall during the LSR reshaping processes, and further affect the final geometrical morphology and optical properties of the generated nanogaps. The discovery of geometry-dependent deformation behaviors of the metallic structures is of great significance to the understanding of the mechanical deformation based nano-manufacturing processes.

Shock-induced plastic deformation of nanopowder Ti during consolidation and spallation.

Consolidating nanopowder metals via impact loading is a potentially significant method for synthesizing and processing bulk nanocrystalline materials. However, until now, the microstructural features, plastic deformation during consolidation, and corresponding mechanisms have been seldom revealed. Using molecular dynamics (MD) simulations, we have studied the plastic deformation, densification, spallation, and micro-jetting in nanopowder titanium (np-Ti) during shock. Upon impact, np-Ti undergoes a transition from heterogeneous plasticity, including basal stacking faults (SFs) and {101̄2} twinning, to homogeneous disordering, as the impact velocity increases. Then the nanopowder structure evolves into a bulk nanostructure after the final densification, contributed by pore collapse. The subsequent detwinning arises during the release and tension stage, conducing to a partial structural recovery. When the impact velocity up ≥ 1.0 km s-1, the spallation is following, prompted via GB-sliding and disordering. Upon shock impact, it also facilitates micro-jetting owing to the presence of nanopores, contributing to the pressure gradient and transverse velocity gradient.

Porosity and specific surface area dependence of shock-induced plasticity and melting in open-cell nanoporous Cu

We systematically study the plasticity and melting behavior in shock loading, as well as their dependence on porosity (ϕ) and specific surface area (γ) for nanoporous copper (NPC), by conducting large-scale non-equilibrium molecular dynamics simulations. During shock compression, the plasticity (i.e., dislocation slips) is dominant at lower impact velocities, while melting is governing at higher impact velocities. With increasing ϕ, both the plasticity and melting undergo the transitions from “heterogeneity” to “homogeneity” along the transverse directions. The increase in γ prompts an apparent heat release and gives rise to the transition from local plasticity to uniform solid disordering at lower impact velocities, while accelerates the melting at higher impact velocities, by converting more surface energy into internal energy. Upon impact, shock-induced pores collapse accelerates the consolidation of NPCs and is controlled by two mechanisms, i.e., the shearing ligament, prompted by plasticity, under low-velocity impact, and the internal micro-jetting facilitated by melting under high-velocity impact.

Effect of shock-induced plastic deformation on mesoscale criticality of 1,3,5-trinitro-1,3,5-triazinane (RDX)

Shock-induced plasticity and structural changes in energetic molecular crystals are well documented. These processes couple with the leading shock wave and affect its propagation, resulting in long, transient responses that are challenging to capture with all-atom simulations due to their time scale. Hence, the effects of this coupling and the transient shock response on the formation of hotspots and the initiation of chemistry remain unclear. To address these challenges, we investigate the role of shock-induced plastic deformation on shock initiation with a recently developed particle-based, coarse-grain model for 1,3,5-trinitro-1,3,5-triazinane (RDX) that utilizes the generalized dissipative particle dynamics with reactions framework. This model enables reactive simulations at micron length scales, which are required to achieve steady-state shock propagation. The simulations show that the shock Hugoniot response of RDX can involve transient behavior for up to 150 ps before steady-state behavior is achieved for shock strengths above the elastic limit. Pore collapse simulations demonstrate that the intensity of the resulting hotspot will weaken as the shock transitions from transient to steady-state behavior, ultimately affecting the shock-to-deflagration transition. Our results highlight the importance of considering the mesoscopic effects of shock-induced plastic deformation in simulations of shock-to-deflagration transitions of high explosives.

Shock-induced melting and crystallization in titanium irradiated by ultrashort laser pulse

Modification of titanium microstructure after propagation of a melting shock wave (SW) generated by a femtosecond laser pulse is investigated experimentally and analyzed using hydrodynamic and atomistic simulations. Scanning and transmission electron microscopy with analysis of microdiffraction is used to determine the microstructure of modified subsurface layers of titanium. We found that two layers are modified beneath the surface. A top surface polycrystalline layer of nanoscale grains is formed from shock-molten material via rapid crystallization. In a deeper subsurface layer, where the shock-induced melting changes into plastic deformation due to attenuation of SW, the grain structure of solid is considerably affected, which results in a grain size distribution differing from that in the intact titanium. Molecular dynamics simulation of single-crystal titanium reveals that the SW front continues to melt even after its temperature drops below the melting curve Tm(P). The enormous shear stress of ∼12 GPa generated in a narrow SW front leads to free slip of atomic planes, collapse of the crystal lattice, and formation of a supercooled metastable melt. Such melt crystallizes in an unloading tail of SW. The mechanical melting ceases after drop in the shear stress giving rise to the shock-induced plastic deformation. The last process triggers a long-term rearrangement of atomic structures in solid. The overall depth of modified layers is limited by SW attenuation to the Hugoniot elastic limit and can reach several micrometers. The obtained results reveal the basic physical mechanisms of surface hardening of metals by ultrashort laser pulses.

Unraveling Anisotropy in Crystalline Orientation under Shock-Induced Dynamic Responses in High-Entropy Alloy Co25Ni25Fe25Al7.5Cu17.5.

Shock-induced plastic deformation and spall damage in the single-crystalline FCC Co25Ni25Fe25Al7.5Cu17.5 high-entropy alloy (HEA) under varying shock intensities were systematically investigated using large-scale molecular dynamics simulations. The study reveals the significant influence of crystalline orientation on the deformation mechanism and spall damage. Specifically, the shock wave velocities in the [110] and [111] directions are significantly higher than that in the [001] direction, resulting in a two-zone elastic-plastic shock wave structure observed in the [110] and [111] samples, while only a single-wave structure is found in the [001] sample. The plastic deformation is dominated by the FCC to BCC transformation following the Bain path and a small amount of stacking faults during the compression stage in the [001] sample, whereas it depends on the stacking faults induced by Shockley dislocation motion in the [110] and [111] samples. The stacking faults and phase transformation in the [001] sample exhibit high reversibility under release effects, while extensive dislocations are present in the [110] and [111] samples after release. Interestingly, tension-strain-induced FCC to BCC phase transformation is observed in the [001] sample during the release stage, resulting in increased spall strength compared to the [110] and [111] samples. The spall strength estimated from both bulk and free surface velocity history shows reasonable consistency. Additionally, the spall strength remains stable with increasing shock intensities. The study discusses in detail the shock wave propagation, microstructure change, and spall damage evolution. Overall, our comprehensive studies provide deep insights into the deformation and fracture mechanisms of Co25Ni25Fe25Al7.5Cu17.5 HEA under shock loading, contributing to a better understanding of dynamic deformation under extreme environments.

Shock-induced plasticity and resultant defect analysis in a vacuum kinetic sprayed silver film

High velocity particle impact generates a shock wave that penetrates the material, resulting in severe plasticity and damage. Finite element method simulation demonstrates that Ag particle momentarily undergoes severe deformation with a high strain rate because of stress fluctuation in the material during the shock wave passage. High-resolution transmission electron microscopy analysis reveals that the lattice was severely distorted, flow-like bent with many geometrically necessary dislocations and stacking faults, twinned at the nanoscale, and collapsed into amorphous state.

Effect of Li element on shocking behavior of Fe-Li alloys

Understanding the effect of alloying elements on shock behaviors is significant for constitutive model of alloy materials at under dynamic high pressure. However, an in-depth understanding about the microscopic mechanism of alloying elements effect on plasticity and phase transition is still limited. In this work, applying non-equilibrium molecular dynamic (NEMD) simulations, shock-induced plasticity and phase transition of Fe-Li alloys was investigated in terms of the compositions of Li, crystallographic direction and shock velocity. The results show that yield stress (or phase change pressure) of Fe-Li alloy was reduced doping Li element, and the large internal stresses caused by concentration of Li element effectively activated the nucleation and multiplication of dislocation loops. But with the increase of Li composition, phase transition is inhibited for shock along [111] direction because of stress release caused by multiplication of dislocation. There is a strong dependence between these microscopic processes and the law of plastic wave propagation. Moreover, with the increase of Li composition, the diversity of phase transition variants was inhibited for shock along [110] direction due to the destruction of initial stress and potential energy uniformity.

Shock-induced dynamic response in single and nanocrystalline high-entropy alloy FeNiCrCoCu

The high-entropy alloys (HEAs) exhibit some unique atomic-scale plastic deformation mechanisms under extreme loading conditions (e.g., shock). Here, four samples are characterized and compared, including nanocrystalline and single-crystal samples in [001], [110], and [111] directions, in terms of their shock-induced dynamic plasticity and failure. Extensive molecular dynamics (MD) simulations show that the anisotropic crystal properties of FeNiCrCoCu HEA significantly affect the Hugoniot elastic limit and the corresponding two-zone elastic-plastic shock wave structures. Importantly, the plastic deformation mechanisms are also tuned by crystal anisotropy. The plastic deformation of the single-crystal sample along [001] direction is dominated by nanotwins and dislocation slip, while those of the other samples only exhibit dislocation slip. The nanotwins are formed by two adjacent stacking faults. In terms of the spallation, the critical shock intensities are different for the four samples. The shock speed threshold of spallation along the [001] and [111] directions is 0.7 km/s, while the [110] direction of 0.6 km/s. Nanocrystalline HEA shows a similar shock speed threshold with the single crystal along [110] direction. The simulation results of spall strength exhibit little discrepancy with the increase of shock velocity. In nanocrystalline HEA, the presence of grain boundaries leads to a low spall strength. During the spall evolution, the voids of single crystals nucleate in the amorphous region formed by the interaction of stacking faults. The spall defects of nanocrystalline samples display a heterogeneous nucleation mechanism at the grain boundaries.

Effect of crystallographic orientations on shock-induced plasticity for CoCrFeMnNi high-entropy alloy

Using nonequilibrium molecular dynamics simulations, we studied the crystallographic-orientation-dependence shock-induced plasticity for the face-centered cubic (FCC) equiatomic CoCrFeMnNi high-entropy alloy (HEA). The local FCC-BCC (body-centered cubic) structural transition was identified by shock loading in the [001] direction, which is attributed to the compression along the <001> direction of original body-centered tetragonal (BCT) structure, and then the BCC structures trend to activate the dislocation nucleation. Further, dislocations slip to produce HCP (hexagonal close-packed) structures in the FCC lattice. In contrast, for the shock loading along the [110] and [111] directions, numerous disordered structures were found due to the insufficient lattice displacement along the <112> direction on the {111} crystal plane of FCC structure and the nonsynergistic behavior of the deformation between the <001>, <110> and <111> directions of BCT structure. In particular, both BCC structures and disordered structures with the Mn-rich composition characteristics play a significant role to promote the development of localized plastic deformation, which is related to the stress concentration around Mn caused by the inherent local inhomogeneity in the CoCrFeMnNi HEA. Moreover, under shock loading, the high atomic fraction of the local BCC structures and disordered structures can further promote chemical composition heterogeneity in these structures. Our results provide some significant insights for understanding the shock-induced plasticity of the CoCrFeMnNi HEA.

Shock-induced plasticity and phase transformation in single crystal magnesium: an interatomic potential and non-equilibrium molecular dynamics simulations

An effective and reliable Finnis–Sinclair (FS) type potential is developed for large-scale molecular dynamics (MD) simulations of plasticity and phase transition of magnesium (Mg) single crystals under high-pressure shock loading. The shock-wave profiles exhibit a split elastic–inelastic wave in the [0001]HCP shock orientation and a three-wave structure in the [10-10]HCP and [-12-10]HCP directions, namely, an elastic precursor, a followed plastic front, and a phase-transition front. The shock Hugoniot of the particle velocity (U p) vs the shock velocity (U s) of Mg single crystals in three shock directions under low shock strength reveals apparent anisotropy, which vanishes with increasing shock strength. For the [0001]HCP shock direction, the amorphization caused by strong atomic strain plays an important role in the phase transition and allows for the phase transition from an isotropic stressed state to the product phase. The reorientation in the shock directions [10-10]HCP and [-12-10]HCP, as the primary plasticity deformation, leads to the compressed hexagonal close-packed (HCP) phase and reduces the phase-transition threshold pressure. The phase-transition pathway in the shock direction [0001]HCP includes a preferential contraction strain along the [0001]HCP direction, a tension along [-12-10]HCP direction, an effective contraction and shear along the [10-10]HCP direction. For the [10-10]HCP and [-12-10]HCP shock directions, the phase-transition pathway consists of two steps: a reorientation and the subsequent transition from the reorientation hexagonal close-packed phase (RHCP) to the body-centered cubic (BCC). The orientation relationships between HCP and BCC are (0001)HCP ⟨-12-10⟩HCP // {110}BCC ⟨001⟩BCC. Due to different slipping directions during the phase transition, three variants of the product phase are observed in the shocked samples, accompanied by three kinds of typical coherent twin-grain boundaries between the variants. The results indicate that the highly concentrated shear stress leads to the crystal lattice instability in the elastic precursor, and the plasticity or the phase transition relaxed the shear stress.

Molecular dynamics study of temperature effects on shock response and plastic deformation mechanism of CoCrFeMnNi high-entropy alloys

High-entropy alloys have broad application prospects in aviation, aerospace, military and other fields due to their excellent mechanical properties. Temperature is an important external factor affecting the shock response of high-entropy alloys. In this paper, we investigate the effects of temperature on the shock response and plastic deformation mechanism of CoCrFeMnNi high-entropy alloys by using molecular dynamics method. The effects of temperature on the atomic volume and the radial distribution function of CoCrFeMnNi high-entropy alloy are studied. Then, the piston method is used to generate shock waves in the sample to study the shock response of CoCrFeMnNi high-entropy alloy. We observe the evolution of atomic-scale defects during the shock compression by the polyhedral template matching method. The results show that the shock pressure, the shock wave propagation velocity, and the rising of shock-induced temperature all decrease with the initial temperature increasing. For example, when piston velocity &lt;i&gt;U&lt;/i&gt;&lt;sub&gt;p&lt;/sub&gt; = 1.5 km/s, the shock pressure at an initial temperature of 1000 K decreases by 6.7% in comparison with that at 1 K. Moreover, the shock Hugoniot elastic limit decreases linearly with the increase of temperature. The Hugoniot &lt;i&gt;U&lt;/i&gt;&lt;sub&gt;p&lt;/sub&gt;-&lt;i&gt;U&lt;/i&gt;&lt;sub&gt;s&lt;/sub&gt; curve of CoCrFeMnNi HEA in the plastic stage can be linearly fitted by the formula &lt;i&gt;U&lt;/i&gt;&lt;sub&gt;s&lt;/sub&gt; = &lt;i&gt;c&lt;/i&gt;&lt;sub&gt;0&lt;/sub&gt; + &lt;i&gt;sU&lt;/i&gt;&lt;sub&gt;p&lt;/sub&gt;, where &lt;i&gt;c&lt;/i&gt;&lt;sub&gt;0&lt;/sub&gt; decreases with temperature increasing. As the shock intensity increases, the CoCrFeMnNi high-entropy alloy undergoes complex plastic deformation, including dislocation slip, phase transformation, deformation twinning, and shock-induced amorphization. At relatively high initial temperature, disordered clusters appear inside CoCrFeMnNi HEA, which together with the BCC (body-centered cubic) structure transformed from FCC (face-centered cubic) and disordered structure are significant dislocation nucleation sources. Compared with other elements, Mn element accounts for the largest proportion (25.4%) in disordered cluster. Owing to the large atomic volume and potential energy, large lattice distortion and local stress occur around the Mn-rich element, which makes a dominant contribution to shock-induced plastic deformation. At high temperatures, the contribution of Fe element to plastic deformation is as important as that of Mn element. The research results are conducive to understanding the shock-induced plasticity and deformation mechanisms of CoCrFeMnNi high-entropy alloys in depth.

Molecular dynamic simulations of plasticity and phase transition in Mg polycrystalline under shock compression

We have investigated the shock-induced plasticity and phase transition in the hexagonal columnar nanocrystalline Mg by large-scale nonequilibrium molecular dynamics simulations. The preexisting grain boundaries (GBs) induce the nucleation of the {10–12} twins for the local stress relaxation. The twins grow up in grains leading to the orientation rotation. The phase transition from the hexagonal close-packed phase to the body-centered cubic phase begins when the migrating twin GBs meet in A- and C-type grains, and continues in the plastic deformation regions. The phase-transition pathway involves two steps: the reorientation and phase transformation.

Unveiling grain size effect on shock-induced plasticity and its underlying mechanisms in nano-polycrystalline Ta

To study macroscopic mechanical response and microscopic deformation mechanism of nano-polycrystalline (NPC) Ta under shock compression, massive-scale non-equilibrium molecular dynamics (NEMD) simulations were systematically performed. The effects of grain size, shock strength on the wave structure, plasticity mechanism and flow stress were unveiled. Under the weak shock compression, as the grain size increases, a transition of plastic wave structure from single to double and then to single one was found, driven by the transition of dominant deformation mechanism from grain boundary (GB)-mediated plasticity to its coexistence with twinning and slipping and then to twinning and slipping, respectively. A transition from twinning-to slipping-preferred plastic deformation as the grain size exceeds ~30 nm was captured in the simulations and explained by a simple theoretical model proposed in this work. Under the strong and ultra-strong shock compressions, twinning-detwinning and amorphization-recrystallization were revealed as the dominant deformation mechanisms, respectively, which show weak grain size dependences. The flow stresses at the Hugoniot state were calculated, which follow the Hall-Petch relation under the weak and strong shocks but show complexity under the ultra-strong shock. These results can help understand intrinsic grain size and extrinsic shock strength effects on the mechanical and microstructural responses of Ta, as a key structural and typical BCC metal.

Shock-induced plasticity in nanocrystalline iron: Large-scale molecular dynamics simulations

Large-scale nonequilibrium molecular dynamics (MD) simulations of shock waves in nanocrystalline iron show evidence of plasticity before the polymorphic transformation takes place. The atomistic structure in the shock direction shows an elastic precursor, plastic deformation, and shock-induced phase transformation from bcc to hcp iron. In this Rapid Communication, large-scale MD models show that the shock response of iron is highly related to the ramp time of the applied shocks. For long ramp times we observe significant plastic relaxation and formation of microstructure defects. Pressure-induced phase transformations in iron are accompanied by stress relaxation achieving almost fully relaxed three-dimensional hydrostatic final states. The evolution of the stress relaxation is in agreement with theory and experiments. Analysis of the x-ray diffraction patterns calculated from the atomistic structure using the Debye equation revealed pronounced anisotropy of the line broadening that is caused by stacking faults in hcp Fe and by dislocations in bcc Fe.

Shock-induced plasticity and damage in single-crystalline Cu at elevated temperatures by molecular dynamics simulations

Initial temperature effects on shock responses, including shock-induced plasticity and spalling damage behaviors of single-crystalline Cu are investigated by molecular dynamics simulations. Firstly, initial temperature effects on stress wave profiles are investigated. The simulations show that shock Hugoniot stress deceases as initial temperature increases, which can be explained by the Rakine-Hugoniot conservation theory. Initial temperature effects on dislocation density are studied. It is found that the dislocation density decreases as initial temperature increases. Shock-induced spalling is dominated by cavitation, i.e., void nucleation, growth and coalescence. Initial temperature effects on cavitation are discussed. In cases of relatively low shock intensity, the total number of voids increases as initial temperature rises; for strong shock intensities that induce melting and micro-spalling, initial temperature effects on the total number of voids are not obvious. Furthermore, initial temperature effects on spall strength are found to be dependent on shock intensity. For relatively weak shock intensity, the simulations show that spall strength starts to drop when initial temperature exceeds 900 K, far below the melting temperature, this result is well consistent with previous experimental measurements; however, for high shock intensity, our simulations predict that spall strength decreases monotonically as initial temperature increases.

Dislocation behavior in nickel and iron during laser shock-induced plastic deformation

Laser shock peening is one of the most effective surface strengthening techniques, which uses laser shock-induced plastic deformation to optimize surface stress state and microstructures of target material. In this paper, dislocation dynamics simulation was used to investigate laser shock induced ultra-high strain rate plastic deformation of face-centered cubic (FCC) nickel and body-centered cubic (BCC) iron. Molecular dynamics was employed to calculate dislocation mobility. Based on the obtained dislocation mobility coefficient, dislocation dynamics models of nickel and iron were established. Results show that the velocity of dislocation motion increases as temperature decreases. Under ultra-high strain rate deformation, dislocation density of nickel increases while dislocation density of iron decreases as temperature rises. Moreover, iron exhibits thermal softening while nickel exhibits thermal hardening under laser shock loading. Plastic deformation dominated by dislocations is sensitive to loading direction, depending on the Schmidt factor of the slip system. The ultra-high strain rate induced by laser shock can effectively increase dislocation density by promoting dislocation multiplication and suppressing dislocation annihilation.

Shock induced plasticity and phase transition in single crystal lead by molecular dynamics simulations

Shock-induced plasticity and phase transition in single crystal lead are investigated by nonequilibrium molecular dynamics simulations. Under dynamic shock loading, the appearance of plasticity in materials precedes that of phase transition. Plasticity mainly causes two effects: one is that plasticity has a significant relaxation effect on shear stress, and the other is that deformation twinning serves as important nucleation sites for the phase transition. This twinning is caused by mutual impediments among different cross-slips and {111} slips. There are three main stages in the dynamic phase transition process of lead: fcc → bcc-like phase transition, plasticity, and hcp phase formation and growth. Moreover, phase transition has a more significant relaxation effect on shear stress, which relaxes the shear stress to a minimum value. The spall strength of lead decreases as the shock intensity increases, but its rate of decrease under different shock intensities is different. Plasticity, especially phase transition, would obviously result in a lower rate of decrease in spall strength.