Plastic Wave Speed Research Articles

Smoothed particle hydrodynamics implementation of the standard viscous–plastic sea-ice model and validation in simple idealized experiments

Abstract. The viscous–plastic (VP) rheology with an elliptical yield curve and normal flow rule is implemented in a Lagrangian modelling framework using the smoothed particle hydrodynamics (SPH) meshfree method. Results show, from a perturbation analysis of SPH sea-ice dynamic equations, that the classical SPH particle density formulation expressed as a function of sea-ice concentration and mean ice thickness leads to incorrect plastic wave speed. We propose a new formulation for particle density that gives a plastic wave speed in line with theory. In all cases, the plastic wave in the SPH framework is dispersive and depends on the smoothing length (i.e., the spatial resolution) and on the SPH kernel employed in contrast to its finite-difference method (FDM) implementation counterpart. The steady-state solution for the simple 1D ridging experiment is in agreement with the analytical solution within an error of 1 %. SPH is also able to simulate a stable upstream ice arch in an idealized domain representing the Nares Strait in a low-wind regime (5.3 m s−1) with an ellipse aspect ratio of 2, an average thickness of 1 m and free-slip boundary conditions in opposition to the FDM implementation that requires higher shear strength to simulate it. In higher-wind regimes (7.5 m s−1) no stable ice arches are simulated – unless the thickness is increased – and the ice arch formation showed no dependence on the size of particles, in contrast to what is observed in the discrete-element framework. Finally, the SPH framework is explicit, can take full advantage of parallel processing capabilities and shows potential for pan-Arctic climate simulations.

Shock Wave Mediation by Closed-Cell Aluminum Foams

Closed-cell aluminum foams of density 0.3 to 0.46 g/cc (1 g/cc=1,000 kg/m3) were exposed to incident shock waves of 10–11 bar (1 bar=1.01325×105 N/m2) pressure in a shock tube to study the shock-induced stress on the end wall, which was measured using a fast-response polyvinylidene difluoride (PVDF) sensor. The experimental results revealed amplification of the end-wall stress by 2.5 to 4 times compared to pressure exerted on the front surface of the foams. Shock compression of foams is discussed based on the stress history output of the PVDF. The properties of the densified foam samples, the particle and plastic wave speeds in foams, based on theory, are presented. A numerical analysis of loading the foams with shocks was conducted using Abaqus software to complement the experimental results. The numerical prediction of the end-wall stress also suggests stress amplification by a factor of 2.6 to 3.4. The information can be useful to designers of shock attenuators and defense shields using aluminum foams.

A fully coupled thermomechanical dislocation model for explosive crystal (β-HMX) subjected to isentropic compression loading

A fully coupled thermomechanical dislocation model is developed to analyze the thermomechanical responses of explosive crystal β-cyclotetramethylene tetranitramine (β-HMX) subjected to isentropic compression loading. The volumetric thermoelastic deformation is considered by a complete equation of state (logarithmic EoS) and the dislocation-based plasticity includes the automatic transition from the low-rate thermally activated to the high-rate drag limited regime. The proposed model is applied for the first time to simulate isentropic compression experiments (ICE) on oriented HMX single crystals. The calculated results can well capture the isentropic elastic limit, stress relaxation and steeper plastic wave speed of the ICE wave profiles. Nonlinear thermoelasticity by both pressure-dependent elasticity tensor and the complete EoS is shown to be responsible for the obvious steepness in wave profiles. The anisotropic elastoviscoplastic response can be distinctly reflected by detailed dislocation activities on seven slip systems instead of phenomenological model just through empirical determination. Pure isentrope is decoupled from quasi-isentrope by subtracting the contribution from deviatoric strain and thermal effect, providing thermodynamically consistent data for complete EoS. Contributions of anisotropic temperature rises by pressure volume work and dislocation-based plasticity work are influenced by loading history. Results provide insights into understanding the transition from isentropic wave to shock wave and ignition behavior of explosives at the mesoscale under complex thermodynamic scenarios.

A new model for the time delay between elastic and plastic wave fronts for shock waves propagating in solids

A time delay is created between elastic and plastic wave fronts because of the difference between the elastic longitudinal sound speed and the plastic shock wave velocity. Over a short propagation distance, the time delay between the elastic and plastic wave fronts at the Hugoniot elastic limit (HEL) is nonlinear, while at larger distances, the time delay is linear. In this work, a new time delay model is introduced that is based on the distance traveled by the waves and using the Rayleigh–Hugoniot jump relations for elastic–perfectly plastic materials. The results of the model have shown in FCC metals the subsonic shock velocity is due to the reduction of shear stress in an unsteady wave being greater than the one in the steady wave. The reduction of the plastic shock wave speed and formation of the elastic shock at the moment of impact are found to result in the nonlinear relationship of the lag between elastic and plastic wave fronts. For calculating the nonlinear time delay in a relaxing material, the lower HEL must be used; the elastic shock is important when the difference between the longitudinal elastic sound speed and the plastic shock wave speed is very small or when the ratio of the HEL to the applied stress is high. In BCC metals, V, Cr, and W, a different behavior has been observed which is in contrast to FCC metals, Ag, Al, and Cu. Therefore, the different behavior is due to a different mechanism that occurs in BCC metals.

The effects of plastic waves on the numerical convergence of the viscous–plastic and elastic–viscous–plastic sea-ice models

The plastic wave speed is derived from the linearized 1-D version of the widely used viscous–plastic (VP) and elastic–viscous–plastic (EVP) sea-ice models. Courant–Friedrichs–Lewy (CFL) conditions are derived using the propagation speed of the wave. 1-D numerical experiments of the VP, EVP and EVP⁎ models successfully recreate a reference solution when the CFL conditions are satisfied, in agreement with the theory presented. The IMplicit–EXplicit (IMEX) method is shown to effectively alleviate the plastic wave CFL constraint on the timestep in the implicitly solved VP model in both 1-D and 2-D. In 2-D, the EVP and EVP⁎ models show first order error in the simulated velocity field when the plastic wave is not resolved. EVP simulations are performed with various advective timestep, number of subcycles, and elastic-wave damping timescales. It is found that increasing the number of subcycles beyond that needed to resolve the elastic wave does not improve the quality of the solution. It is found that reducing the elastic wave damping timescale reduces the spatial extent of first order errors cause by the unresolved plastic wave. Reducing the advective timestep so that the plastic wave is resolved also reduces the velocity error in terms of magnitude and spatial extent. However, the parameter set required for convergence to within the error bars of satellite (RGPS) deformation fields is impractical for use in climate model simulations. The behavior of the EVP⁎ method is analogous to that of the EVP method except that it is not possible to reduce the damping timescale with α=β.

Influence of prestress on shock behavior of polycrystalline iron via atomistic simulations

Plasticity behavior and phase transition of metal Fe subjected to shock loading have attracted considerable attention in shock physics community, in particular for underlying relationship between them. Experimental examinations and atomistic simulations on shocked Fe have displayed a three-wave structure: elastic wave, plastic wave and transformation wave. However, these studies are primarily limited to the one-dimensional planar case. Recently, owing to the rapid development of experimental techniques, investigating dynamic property of shocked metal has extended to the multi-dimensional loading conditions, such as cylindrical or spherical shocks. In this regard, fruitful findings are achieved, for example, twinning ratio in polycrystalline Fe under implosive compression is found to be much higher than that under planar shock, implying that the the complex stress state plays a critical role. In this paper, we explore the effects of prestress on plasticity and phase transition of shocked polycrystalline iron. The imposed presstress normal to the impact direction in one-dimensional planar shocking represents the varying deviatoric stress, and does not nearly affect the principal stress. The utilized empirical potential for iron could describe the plasticity dislocation and phase transition very well. The simulations show that as the prestress increases, the shock speed at elastic stage and Hugoniot elastic limit increase, which is in accordance with the theoretical analyses based on shock wave theory and experimental measurement. Meanwhile the plastic wave speed increases more quickly and catches up with the transformation wave more easily, resulting in a steep shockwave front. Atomistic snapshots show that plasticity dislocation stemming from the grain boundary precedes phase transition, where most of BCC atoms are transformed into the HCP atoms and shear stress significantly decreases. Further observations from these images find that plastic zone becomes narrower with increasing prestress, representing a shorter plastic relaxation time, which accelerates the completion of phase transition. This rapid phase transition process is also indicated by quantitatively evaluating the ratio of transitioned closed packed atoms as a function of evolution time. The origin based on the atomistical prediction model of Fe phase transition is attributed to the fact that higher prestress gives rise to the larger von-Mises stress for easier dislocation emission while lower one cannot. But the final transformed atoms are independent of prestress. Additionally, the measured free surface velocity profiles from planar and cylindrical impact loading validate the simulations conducted here. These findings will help to understand experimentally the microscopically dynamic evolution of Fe, imposed by complex stress state.

NONUNIQUENESS AND INSTABILITY OF CLASSICAL FORMULATIONS OF NONASSOCIATED PLASTICITY, I: CASE STUDY

A dynamic instability, which is relatively unexplored in the literature despite having been long ago previously asserted to exist for any conventional nonassociated plastic flow model, is illustrated by means of an example problem. This instability is related to a condition known as achronicity, in which the wave speed in plastic loading is greater than that in elastic unloading (making it absolutely not related to the well-studied phenomena of localization and flutter). The one-dimensional example problem initializes an elastic-plastic half-space to a initially quiescent uniform state of prestress that places it in an achronic condition if a nonassociated flow rule is used. The initial stress state is perturbed by an axial stress pulse applied at the surface. The problem is first solved analytically for the case of constant wave speeds, and it is shown to possess a two-parameter family of nonunique solutions. These solutions are unstable in that both the amplitude and the width of the propagating pulse increase linearly with time. The casestudy problem technically represents spontaneous motion from a quiescent state, but it does not violate thermodynamics (as the energy driving the instability is available from elastic stored energy of the initial prestress). The example problem is additionally solved numerically for both constant and nonconstant plastic wave speeds, where the instability is observed in either case. Furthermore, it is shown that neither the constant nor the nonconstant wave speed solution converges with mesh refinement, which therefore represents a numerical inadmissibility associated with the underlying loss of uniqueness of solution. It is the nonuniqueness of the unstable solution, not the existence of the instability itself, that is of primary concern. Unlike conventional localization instability, this achronic instability is not yet known to be a real phenomenon. This case-study problem illustrates the need for novel laboratory testing methods sufficient to determine if the instability is truly physical, or merely an anomalous shortcoming of classical nonassociated plasticity formulations. Some guidance for appropriate laboratory testing is presented, with emphasis on why such testing is highly nontrivial as a result of irreducible uncertainty in direct validation of a regular flow rule.

Mathematical modelling of elastoplasticity at high stress

This study describes a simple mathematical model for one-dimensional elastoplastic wave propagation in a metal in the regime where the applied stress greatly exceeds the yield stress. Attention is focused on the increasing ductility that occurs in the over-driven limit when the plastic wave speed approaches the elastic wave speed. Our model predicts that a plastic compression wave is unable to travel faster than the elastic wave speed, and instead splits into a compressive elastoplastic shock followed by a plastic expansion wave.

Ultra High-Speed Cutting Experiment under the Cutting Condition that Cutting Speed Exceeds Plastic Wave Speed of Workpiece

Raising a cutting speed to above speed of a plastic wave of a workpiece material induces the high levels of the hydrostatic stresses in the shear zone, because a plastic wave traveling there becomes a shock wave. In order to ascertain the cutting phenomena occurring under the ultra high-speed cutting condition, the cutting experiments of a pure lead with cutting speeds of up to 140 m/s are performed with a high-speed impact cutting tester developed. The experimental result reveals that the cutting mechanism, especially chip formation, changes remarkably and the friction angle at the tool-chip interface rises in the ultra high-speed cutting. It can be explained that these phenomena arise from the plastic shock wave in the shear zone.

On the plastic wave speeds in rate-independent elastic–plastic materials with anisotropic elasticity

This paper presents a theoretical study of the speeds of plastic waves in rate-independent elastic–plastic materials with anisotropic elasticity. It is shown that for a given propagation direction the plastic wave speeds are equal to or lower than the corresponding elastic speeds, and a simple expression is provided for the bound on the difference between the elastic and the plastic wave speeds. The bound is given as a function of the plastic modulus and the magnitude of a vector defined by the current stress state and the propagation direction. For elastic–plastic materials with cubic symmetry and with tetragonal symmetry, the upper and lower bounds on the plastic wave speeds are obtained without numerically solving an eigenvalue problem. Numerical examples of materials with cubic symmetry (copper) and with tetragonal symmetry (tin) are presented as a validation of the proposed bounds. The lower bound proposed here on the minimum plastic wave speed may also be used as an efficient alternative to the bifurcation analysis at early stages of plastic deformation for the determination of the loss of ellipticity.

Development of Orthogonal Impact Cutting Testing Machine

FEM cutting simulation predicts that the plastic shock waves are generated that develop the high levels of hydrostatic stress in the shear zone when cutting speed exceeds the plastic wave speeds of the workpiece material. The orthogonal impact cutting testing machine was developed to confirm this phenomenon experimentally. In the testing machine, two guide rails are set up in parallel. The cutting tool and the workpiece are installed on the blocks that slide on the rail. Each block connected with the piston in the air tube by the pushrod is launched by the expansion of compressed air, and is accelerated rapidly. When the two blocks passes each other, cutting is done. This paper describes the details of the developed impact cutting testing machine and experimental results of cutting a pure lead at cutting speed up to 65m/s.

Dynamic buckling of elastic–plastic square tubes under axial impact—I: stress wave propagation phenomenon

The speeds of the stress waves that can propagate in an elastic–plastic medium with isotropic linear strain hardening in a plane stress state are obtained in order to analyse the influence of the transient deformation process on the initiation of buckling in square tubes under axial impact. The kinematic conditions across a surface of discontinuity were employed to obtain the stress wave propagation speeds for an initial condition [ v x ]= V 0. It is shown that the plastic wave speeds depend on the stress state and on the direction of wave propagation. The material hardening properties have a stronger effect on the speed of the slow plastic wave, while the shear stress affects both the speeds of the fast and slow plastic waves. The magnitude of the instantaneously applied in-plane load [ σ xx ] at t=0, which results from an impact in the direction (n x=1, n y=0) , is obtained as a function of the initial velocity V 0 when using the differential relationships along the characteristics. A numerical simulation of an in-plane impact on a finite plate by a large mass with an initial velocity shows that the predicted stress wave speeds from the theoretical analysis agree with the speed of the propagation of the plastic zone in the plate. It is shown that the initial buckling pattern forms within a sustained axial plastic flow and remains unchanged when large displacements develop. Therefore, it is concluded that the transient process in elastic–plastic plates subjected to an in-plane impact plays a significant role for the formation of the final buckling shape and this process can be analysed using estimates for the stress wave speeds obtained for an elastic–plastic medium in a plane stress state.

Time resolved plastic deformations in rods subjected to transverse impact

This paper describes experiments to record deformation profiles versus time in metallic rods subjected to transverse ballistic impacts. Rod materials included a tungsten alloy, four alloys of steel, titanium, and copper. The deformation profiles were analyzed to determine time resolved transverse plastic deformation wave speeds in all the rod materials and to estimate effective fracture shear strains in some of the materials. These material parameters are key elements in an analytical model to estimate deformations and failure in long rod penetrators subjected to transverse loads arising from non-ideal plate penetration geometries (yaw and obliquity). The rod materials were also subjected to tensile tests and Taylor anvil impact tests to measure static tensile and dynamic compressive strengths and longitudinal plastic wave speeds. The Taylor anvil and transverse rod impact tests together furnish the required dynamic material properties for comparing the penetration effectiveness of candidate rod materials and geometries.

Effects of elastic anisotropy on strain localization andflutter instability in plastic solids

The influence of elastic anisotropy on elastic–plastic acceleration wave speeds isanalyzed when the elastic anisotropy is described by a second-order fabric tensor. For aparticular—and physically sound—structure of the elastic tensor, a correspondence principle isestablished that reduces the analysis of strain localization to a problem relative to elasticisotropy. Thus analytic solutions available in the framework of elastic–plastic constitutive lawswith isotropic elasticity can be used. Applications show the strong effects of elastic anisotropy onall the localization characteristics.For non-associative flow-laws, flutter instability,corresponding to the occurrence of complex conjugate squares of wave speeds, is considered. Itis shown that, while this instability is precluded for elastic–plastic solids with elastic isotropyand obeying deviatoric associativity, this exclusion property is not preserved in the presence ofanisotropic elasticity. In fact, we exhibit a particular structure of elasticity for which aninfinitesimal amount of anisotropy is shown to give rise to flutter.

A numerical scheme for extracting strength model coefficients from Taylor test data

A simple pair of ordinary differential equations describing a one-dimensional model of a deforming Taylor impact specimen were numerically integrated. The Johnson-Cook strength model was assumed to apply and was used to estimate material strengths and plastic wave speeds during the deformation process. The material constants of the Johnson-Cook model were adjusted using an optimizer so that experimentally measured deformed geometries were matched with those calculated by the integrator. In this fashion Taylor test data were used to back-out strength model coefficients. Event durations, and peak values of stresses, equivalent plastic strains, equivalent plastic strain rates, and temperatures were also obtained from the simple numerical model. The EPIC finite element code was used to investigate the validity of the predictions of the simple model. Three different materials are considered (OFE copper, 6061-T6 aluminum, and 4340 steel) and the results obtained are reasonable.

A continuum mechanics code analysis of steady plastic wave propagation in the Taylor test

Simple conservation relationships (jump conditions) in conjunction with postulated material constitutive behavior are applied to steady plastic strain waves propagating in problems of uniaxial stress and Taylor Cylinder Impact. These problems are simulated with a two-dimensional Lagrangian continuum mechanics code for the purpose of numerically validating the jump relationships as an accurate analytical representation of plastic wave propagation. The constitutive behavior used in this effort assumes isotropy and models the thermodynamic response with a Mie-Grunisen Equation-of-State and the mechanical response with the rate-dependent Johnson-Cook and MTS flow stress models. The jump relationships successfully replicate the results produced by continuum code simulations of plastic wave propagation and provide a methodology for constructing mechanical constitutive models from experimental plastic wave speed information. Comparisons are also presented between experimental speeds from Taylor Cylinder Impact tests with jump relationships and continuum code predictions, indicating that the above mentioned flow stress models may not accurately capture plastic wave propagation speeds in annealed and hardened copper.

Generalized Method of Separation of Variables on the Nonlinear Elastic-Plastic Wave Motion

The method of separation of variables is developed in addition to the Karman method and the method of characteristics for the wave motion of uniaxial stress in rods. Rate-independent theory is considered and it is shown that the plastic wave speed is independent of the constant n in the stress-strain law. Stresses are determined in explicit forms for two cases.

Fracture mechanical modeling of brittle fracture propagation and arrest of steel (1)

A model is proposed which simulates brittle fracture propagation and arrest of steel with finite thickness. Based on the critical stress criterion for brittle fracture and on stress field solution for dynamically running crack in an elastic-plastic solid, dynamic fracture toughness (KD) is calculated as a function of temperature and crack velocity. The calculated KD once decreases and then increases with increasing crack velocity. Upper limit crack velocity coincides with the plastic surface wave speed and decreases with increasing temperature. The predicted dependencies of KD on temperature and crack velocity agree with previously established experimental data. Effects of side-ligament formed near the plate surface and crack-tunneling at the inside of the plate are also taken into account for simulating the brittle crack propagation. The model predicts crack velocity, shear lip thickness, aspect ratio of the tunneling crack from the fundamental mechanical properties of steel. The model also predicts a lower bound value of applied stress intensity factor and crack velocity, the former of which may correspond to the arrest toughness, Kca.

Influence of non-classical elastic-plastic constitutive features on shock wave existence and spectral solutions

T he influence of non-classical elastic-plastic constitutive features on dynamically moving discontinuities in stress, strain and material velocity is investigated. Non-classical behavior here includes non-normality of the plastic strain increment to the yield surface, plastic compressibility, pressure sensitivity of yield and dependence of the elastic moduli on plastic strain. D rugan and S hen's (1987) analysis of dynamically moving discontinuities with strain as well as stress jumps in classical materials is shown to be valid for a broad class of non-associative material models until deviation from normality exceeds a critical (non-infinitesimal) level. For these non-classical materials, an inequality that bounds the magnitude of the stress jump is derived, which is information not obtainable from a standard spectral analysis of a shock. For the special case of stress discontinuities with continuous strain or for quasi-static deformations, this inequality is shown to rule out jumps in specific projections of the stress tensor unless the non-normality is sufficiently large. These results invalidate a recent claim in the literature that an infinitesimal amount of non-normality permits moving surfaces of discontinuity in stress (with no strain jump) near the tip of a dynamically advancing crack tip. Using a very general plastic constitutive law that subsumes most non-classical (and classical) descriptions currently in use, a complete closed form solution is obtained for the plastic wave speeds and eigenvectors. A novel feature of the analysis is the clarity and completeness of the solutions. If the elastic part of the response is isotropic, one plastic wave speed equals the elastic shear wave speed, while the other two possible wave speeds depend in general on the stress and plastic strain within the shock transition layer. Concise necessary and sufficient conditions for real eigenvalues and for vanishing eigenvalues are derived. The real eigenvalues are classified by numerical sign and ordering relative to the elastic eigenvalues. The geometric multiplicity of plastic eigenvectors associated with elastic eigenvalues is shown to depend on the stress state within the shock transition layer. These solutions, several of which hold for arbitrary elastic anisotropy, are also applicable to acceleration waves and localization problems and to materials with dependence of the elastic moduli on plastic strain. Such elastic-plastic coupling is shown to imply a non-self-adjoint fourth order tangent stiffness tensor even if the plastic constitutive law is associative.

A One-Dimensional, Two-Phase Flow Model for Taylor Impact Specimens

In this paper, a simple theoretical analysis of an old problem is presented. The analysis is more complete than earlier versions, but retains the mathematical simplicity of the earlier versions. The major thrust is to separate the material response into two phases. The first phase is dominated by strain rate effects and has a variable plastic wave speed. The second phase is dominated by strain hardening effects and has a constant plastic wave speed. Estimates for dynamic yield stress, strain, strainrate, and plastic wave speed during both phases are given. Comparisons with several experiments on OFHC copper are included.