Elastic-plastic Wave Research Articles

Piecewise oblique boundary treatment for the elastic–plastic wave equation on a cartesian grid

Numerical schemes for hyperbolic conservation laws in 2-D on a Cartesian grid usually have the advantage of being easy to implement and showing good computational performances, without allowing the simulation of “real-world” problems on arbitrarily shaped domains. In this paper a numerical treatment of boundary conditions for the elastic–plastic wave equation is developed, which allows the simulation of problems on an arbitrarily shaped physical domain surrounded by a piece-wise smooth boundary curve, but using a PDE solver on a rectangular Cartesian grid with the afore-mentioned advantages.

Plasticity effects in dynamically loaded nickel aluminide bicrystals

This work is concerned with achieving an improved understanding of the role of material strength effects during interactions of shock waves with grain boundaries. To this end, experiments have been performed using nanosecond laser shocks of nickel aluminide bicrystals at tens of GPa. Velocity histories were measured along a line on the free surface of the bicrystals and used to characterize the material behavior. Unstable plastic flow in 〈 1 0 0 〉 grains was seen to occur when loaded above 700 m s −1 free surface velocity. Observations of the grain boundary region showed that a smooth transition occurred between the elastic precursors in both grains as well as the plastic waves (when plastic flow is evident). Futhermore, the length of the transition zone across the boundary was observed to be larger than the projection of the boundary, indicating that refraction of the incident wave resulted in the initiation of transmitted and reflected waves. A model is developed to explain the experimentally observed behavior. The model describes the scattering of elastic–plastic waves off of grain boundaries in crystalline materials where slip is the active deformation mechanism. Reflected elastic release waves and transmitted plastic waves scattering at large angles (i.e. 56° for the transmitted wave) from the boundary are predicted for the case of an incident plastic wave propagating along 〈 1 1 1 〉 and impacting a 45° inclined boundary, which appears to be consistent with the experimental VISAR records. The case of an incident 〈 1 0 0 〉 plastic wave could not be treated since the incident polarization vector is either parallel or perpendicular to the operative slip vectors in NiAl.

Instrumented anvil-on-rod impact experiments for validating constitutive strength model for simulating transient dynamic deformation response of metals

Instrumented anvil-on-rod impact experiments were performed to access the applicability of this approach for validating a constitutive strength model for dynamic, transient-state deformation and elastic–plastic wave interactions in vanadium, 21-6-9 stainless steel, titanium, and Ti–6Al–4V. In addition to soft-catching the impacted rod-shaped samples, their transient deformation states were captured by high-speed imaging, and velocity interferometry was used to record the sample back (free) surface velocity and monitor elastic–plastic wave interactions. Simulations utilizing AUTODYN-2D hydrocode with Steinberg–Guinan constitutive equation were used to generate simulated free surface velocity traces and final/transient deformation profiles for comparisons with experiments. The simulations were observed to under-predict the radial strain for bcc vanadium and fcc steel, but over-predict the radial strain for hcp titanium and Ti–6Al–4V. The correlations illustrate the applicability of the instrumented anvil-on-rod impact test as a method for providing robust model validation based on the entire deformation event, and not just the final deformed state.

High-strain-rate dynamic mechanical behavior of a bulk metallic glass composite

The high-strain-rate mechanical properties, deformation mechanisms, and fracture characteristics of a bulk metallic glass (BMG)-matrix composite, consisting of an amorphous Zr57Nb5Cu15.4Ni12.6Al10 (LM106) matrix with crystalline tungsten reinforcement particles, were investigated using gas gun anvil-on-rod impact experiments instrumented with velocity interferometry (VISAR) and high-speed digital photography. The time-resolved elastic-plastic wave propagation response obtained through VISAR and the transient deformation states captured with the camera provided information about dynamic strength and deformation modes of the composite. Comparison of experimental measurements with AUTODYN-simulated transient deformation profiles and free surface velocity traces allowed for validation of the pressure-hardening Drucker–Prager model, which was used to describe the deformation response of the composite. The impacted specimens recovered for post-impact microstructural analysis provided further information about the mechanisms of dynamic deformation and fracture characteristics. The overall results from experiments and modeling revealed a strain to failure of ∼45% along the length and ∼7% in area, and the fracture initiation stress was found to decrease with increasing impact velocity because of the negative strain-rate sensitivity of the BMG.

Energy accumulation and dissipation in metals as a result of structural-scaling transitions in a mesodefect ensemble

The localization of plastic deformation and its wave-like propagation are well-known and experimentally studied phenomena. Modern experiments carried out with highly sensitive infrared cameras have allowed us to investigate in detail energy accumulation and dissipation in the material which accompany the above process. It is found that plastic deformation occurs together with great energy accumulation in the material. The energy accumulation rate peaks on the initial stage of plastic deformation and vanishes at developed plastic flow. At the moment of elastic-plastic transition dissipative waves induced by plastic deformation localization can be generated and propagate in some metals. The given paper puts forward a theoretical model of energy accumulation and dissipation in metals under plastic deformation. The model is based on a statistical description of evolution of an ensemble of typical mesodefects (microshears). The derived constitutive equations adequately describe certain experimentally observed peculiarities of energy dissipation in metals.

Dynamic Mechanical Behavior Characterization of Epoxy-Cast Al + Fe2O3 Thermite Mixture Composites

The dynamic mechanical behavior characterization of epoxy-cast stoichiometric mixtures of nano- or micron-scale aluminum and hematite (Fe2O3) powders is investigated in this work. Experiments conducted on rod-shaped samples, using instrumented reverse Taylor impact tests employing high-speed imaging and velocity interferometry, show that these composites exhibit viscoelastic deformation and brittle fracture behaviors. Upon impact, the samples display significant elastic and plastic deformation during both the loading and unloading stages, as determined from quantitative high-speed camera measurements of the transient deformation states. Approximately 50 pct elastic recovery of total axial strain was observed to occur rapidly (within tens of microseconds) after impact. A one-dimensional elastic-plastic wave propagation analysis was used for estimating the composite’s dynamic average yield stress and total plastic strain. The results reveal that the nano-Al + Fe2O3-containing epoxy composite is most resilient, has the highest strength, and is more capable of absorbing impact energy. The analysis additionally provides detailed information about elastic and plastic wave interactions for discrete times, up to the final state of the material. Calculations and observations through the coupling of high-speed camera images and velocity interferometry (VISAR) measurements show that the elastic recovery coincides with peak axial strain and the interaction of elastic and plastic waves propagating within the rod-shaped specimen. Hence, such an instrumented Taylor test provides a detailed view of the general wave structure within the material upon impact and, at the same time, enables a complete description of the stress-strain response.

Elastic-plastic shock wave profiles in oriented single crystals of cyclotrimethylene trinitramine (RDX) at 2.25GPa

Plate impact experiments were performed on oriented single crystals of the energetic material cyclotrimethylene trinitramine (RDX). The experiments were performed to determine the anisotropic dynamic yield point for the RDX crystal, as well as to provide data for continuum modeling efforts. Impact was on the (111), (210), and (100) planes to access 3, 2, and 0 slip systems, respectively. Velocity history profiles were measured using Doppler interferometry. Impacts on the (210) plane resulted in nominally conventional results, with distinct elastic and plastic waves, stress relaxation, elastic precursor decay, and increasing wave separation with propagation distance. Velocity profiles from impacts on the (111) plane had no discernable precursor, although an inflection seen in the thicker samples might be the nearly overdriven elastic wave. Wave arrival times signaled a slower elastic wave speed in the (111) profiles. Several unexpected features were observed in the elastic precursor of the profiles from impacts on the (100) plane. Up to three distinct step features were resolved in these profiles in the region of the elastic precursor; these features are not understood. In preparing samples for these experiments, it was noted that the (100) crystal slabs were exceptionally brittle. Wave speeds determined from the shock experiments were consistent with both pulse-echo wave speed measurements and wave speeds calculated from the measured elastic tensor. The elastic limit, as indicated by the peak of the leading wave, was relatively isotropic.

Dynamic buckling of elastic-plastic column impacted by rigid body

The dynamic buckling of an elastic-plastic column subjected to an axial impact by a rigid body was discussed by using the energy law. The traveling process of elastic-plastic waves under impact action was analyzed by characteristics method. The equation of lateral disturbance used to analyze the problem was developed by taking into account the effect of elastic-plastic stress wave. The power series solution of this problem has been the power series approach. The buckling criterion of this problem was proposed by analyzing the characteristics of the solution. The relationship among critical velocity and impact mass, critical buckling length, hardening modulus was given by using theoretical analysis and numerical computation.

Analysis of inertia and scale effects on dynamic neck formation during tension of sheet steel

It is well known that a specimen for impact testing must be optimized in terms of its dimensions. The main reason is to reduce strain gradients due to the effects of elastic–plastic wave propagation. On the other hand, when a split Hopkinson bar in tension is applied, the net displacement of the specimen ends is very limited, usually from 2.0 to 3.0 mm. Thus, to reach a maximum strain of 0.5 the specimen length must be reduced in dimensions from 4.0 to 6.0 mm. Consequently, small diameters or lateral dimensions and lengths must be applied to assure one-dimensional deformation. Such small lengths substantially perturb the determination of real material behavior. So the main motivation of this study was to perform a systematic analysis, numerical and analytical, to find differences in the behavior of short and long specimens loaded in impact tension. The finite element code ABAQUS/Explicit has been used to simulate several specimen lengths from 10 to 40 mm submitted to impact velocities ranging from 10 to 100 m/s.

Nonlinear Stability Analysis for the Method of Transport for the Elastic-Plastic Wave Equation

In this paper we present an analytic assessment of the stability and convergence of the so-called method of transport for solving the elastic-plastic wave equation, which is a nonlinear hyperbolic partial differential equation. For the purely elastic wave equation, which is a linear hyperbolic conservation law, one can use von Neumann analysis for proving that stability and convergence immediately follow from the Lax equivalence theorem. For plastic deformation, however, nonlinear stability proofs require greater effort.

HIGH-RESOLUTION SIMULATION OF THE ELASTIC-PLASTIC WAVE EQUATION IN THREE SPACE DIMENSIONS

International Journal of Computational Engineering ScienceVol. 05, No. 03, pp. 619-638 (2004) No AccessHIGH-RESOLUTION SIMULATION OF THE ELASTIC-PLASTIC WAVE EQUATION IN THREE SPACE DIMENSIONSGUIDO GIESEGUIDO GIESESeminar for Applied Mathematics, ETH Zurich, ETH Zentrum, 8004-Zurich, Switzerland Search for more papers by this author https://doi.org/10.1142/S1465876304002599Cited by:0 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail AbstractIn this paper we present an efficient numerical one-step method of high order in space and time for solving the elastic-plastic wave equation in three space dimensions. The basic idea is to decompose the hyperbolic PDE into advection equations, which can be solved numerically, Furthermore, the occurrence of plasticity makes it necessary to solve an ODE for the stress-strain relationship at every point.Keywords:Hyperbolic conservation lawselastic-plastic wave equationhigh order finite-volume schemes References P. Colella, SIAM J. Comp. Phys. 87, 171 (1990). Crossref, Google Scholar M. Fey , R. Jeltsch and A.-T. Morel , Numerical Analysis 1995 , eds. D. Griffiths and G. A. Watson ( Longman , 1996 ) . Google Scholar D. Kröner , Numerical Schemes for Conservation Laws ( Wiley Teubner , 1997 ) . Google Scholar R. J. LeVeque , Numerical Methods for Conservation Laws , Lectures in Mathematics ( Birkhäuser , 1992 ) . Crossref, Google Scholar D. Serre , Systems of conservation laws ( Cambridge University Press , 1999 ) . Crossref, Google Scholar X. Lin , Numerical Computation of stress waves in solids ( Akademie Verlag , 1996 ) . Google ScholarX. Lin and J. Ballmann, Quart. Appl. Math LIII(2), 383 (1995). Google ScholarX. Lin and J. Ballmann, Wave Motion 21, 115 (1995). Crossref, Google ScholarX. Lin and J. Ballmann, ZAMM 75, 267 (1995). Google ScholarX. Lin and J. Ballmann, Nonlinear Waves in Solids, eds. J. L. Wegner and F. R. Norwood (ASME book, 1995) pp. 155–160. Google ScholarX. Lin and J. Ballmann, Journal de Physique III(4), (1994). Google ScholarM. Fey, Journal of Computational Physics 143, 159 (1998). Crossref, Google ScholarM. Fey, Journal of Computational Physics 143, 181 (1998). Crossref, Google Scholar M. Fey, R. Jeltsch, J. Maurer, A.-T. Morel. The method of transport for nonlinear systems of hyperbolic conservation laws in several space dimensions, Proceedings of the Conference on Numerical Analysis, Zeist, 1996 . Google Scholar L. Prandtl. Proceedings of the First International Congress for Applied Mechanics. Delft, 1924, p. 43 . Google Scholar P. L. Gould , Introduction to Linear Elasticity ( Springer Verlag , 1983 ) . Crossref, Google Scholar J. Lubliner , Plasticity Theory ( Macmillan Publishing Company, cop. , 1990 ) . Google Scholar W. K. Nowacki , Stress waves in non-elastic solids ( Pergamon Press , Oxford , 1978 ) . Google Scholar K. W. Morton and R. D. Richtmyer , Difference Methods for Initial-Value Problems ( Krieger Publishing Company , 1995 ) . Google Scholar FiguresReferencesRelatedDetails Recommended Vol. 05, No. 03 Metrics History KeywordsHyperbolic conservation lawselastic-plastic wave equationhigh order finite-volume schemesPDF download

A dynamic high-resolution simulation of 2-D elastic-plastic crack problems

In this paper we present a numerical method for performing higher-order simulations of elastic-plastic waves in solids and we illustrate the accuracy of our approach by various numerical simulations of wave phenomena around cracks. The simulation of crack problems in solids is especially challenging, since the singularities in physical variables occurring at the crack are particularly difficult for numerical schemes to simulate, but they create many interesting physical effects.

A high-resolution residual splitting scheme for the elastic–plastic wave equation

In this paper we present a multi-dimensional scheme for solving the elastic–plastic wave equation that is based on a so-called residual splitting approach. The basic idea of this methodology is to solve the analytic time-derivative of a PDE rather than the PDE itself and then integrate the solution in time numerically. We illustrate the advantages of such an approach with theoretic arguments and practical examples.

Elastic–plastic wave profiles in cyclotetramethylene tetranitramine crystals

The explosive molecular crystal cyclotetramethylene tetranitramine was studied in three orientations in a set of plate impact experiments; the orientations studied were {110}, {011}, and {010} in P21/n space group. The elastic–plastic shock response was measured using laser interferometry. The measured particle velocity profiles showed elastic precursor decay typical of a stress relaxing material. There is anisotropy in elastic shock strength and decay. The amount of precursor decay with propagation distance and stress relaxation behind the elastic shock varied among the orientations. The {010} orientation had larger elastic precursors than did the other two orientations; the {010} crystal does not have the regular plastic deformation mechanisms available to it. Elastic Hugoniots were obtained from the measurements. The inelastic deformation mechanisms may vary with orientation.

Laser shock processing of Ti-6Al-4V alloy

Laser shock processing of Ti-6Al-4V alloy is considered. A mathematical model governing the laser non-conduction-limited heating is introduced and a numerical solution for the temperature distribution is presented. The pressure (recoil pressure) generated across the solid-vapour interface is determined and it is considered as an impact load for the elastic-plastic wave generation in the surface region of the substrate material. The depth of the plastic region is predicted from the elastic-plastic wave propagation analysis. An experiment is carried out to ablate the surface using a neodymium-doped yttrium aluminium garnet laser with 8 ns pulse length and 0.5 J energy content. Scanning electron microscopy is carried out to examine the surface morphology of the irradiated region. It is found that microhardness increases by 1.5 times the base hardness of the substrate material. The depth of the plastic region predicted extends 0.195 mm from the surface, which agrees well with the experimental findings. In addition, some localized microcracks are observed at the surface of the workpiece.

Predictions for weak mechanical ignition of strain hardened granular explosive

Predictions are given for the coupled bulk and grain scale response of initially unstressed, strain hardened granular HMX (C4H8N8O8) due to mild piston impact (impact speeds <100 m/s). Importantly, this response depends on the material’s strain history as the stress necessary for bulk inelastic compaction (crush up) increases with the solid volume fraction. Although the quasistatic compaction behavior of HMX is well characterized, the influence of strain history on the bulk and grain scale dynamic loading response has largely been unexplored. In this study, the initial solid volume fraction of the unstressed material is varied over the range of φf⩽φ0⩽1, where φf=0.655 is its free pour value. A Hugoniot analysis for the bulk material identifies three dispersed compaction wave structures that depend on the impact speed and initial solid volume fraction, and are analogous to elastic-plastic waves in dynamically loaded solids. For increasing impact speed, these structures consist of (1) a single viscoelastic wave; (2) a leading viscoelastic wave and a trailing viscoplastic wave; and (3) a single viscoplastic wave. It is shown that the resulting localized heating near intergranular contact surfaces can trigger sustained combustion of the material. Predictions for the grain scale thermochemical response indicate that significant bulk viscoplastic heating is required for ignition of materials with φf⩽φ0⩽0.88, whereas bulk viscoelastic heating leads to the ignition of denser materials (φ0>0.91). Both viscoelastic and viscoplastic heating are predicted to be important for ignition of materials having 0.88⩽φ0⩽0.91. Within this transition range there is predicted a sharp increase in impact sensitivity as the power input needed for ignition rapidly decreases to a value close to that for the free pour density (0.40 MW/cm2) before increasing again. This result is important for assessing the impact sensitivity and deflagration-to-detonation transition of damaged explosives.

Investigation Into Laser Shock Processing

Laser shock processing is a good candidate for surface industry due to its rapid processing, localized ablation, and precision of operation. In the current study, laser shock processing of steel was considered. The numerical solutions for temperature rise and recoil pressure development across the interface of the ablating front and solid are presented. The propagation of elastic-plastic waves in the solid due to recoil pressure loading at the surface is analyzed and numerical solution for the wave propagation was obtained. An experiment was conducted to ablate the steel surfaces for shock processing. Scanning electron microscopy was carried out to examine the ablated surfaces shock processing while transmission electron microscopy was conducted to obtain dislocation densities after the shock processing. It was found that surface hardness of the workpiece increased in the order of 1.8 times of the base material hardness, and the dislocation was the main source of the shock hardening in the region affected by laser shock processing.

Impact shear-numerical analyses of ASB evolution and failure for Ti-6AI-4V alloy

In this contribution a study is reported on a shear zone evolution at high strain rates for Ti-6Al-4V-alloy. Such process of deformation, leading to development of Adiabatic Shear Band (ASB), is typical for fast and impact shearing. A very special experimental technique developed at LPMM-Metz has permitted for shear tests at a wide range of strain rates and also for development of an advanced constitutive relation valid for that range of strain rates and also for a range of temperatures (range of strain rates from 10 -4 l/s to about 10 5 l/s). This experimental technique called the Modified Double Shear has been applied to test Ti-6Al-4V alloy. In order to estimate the effect of stress concentration on the sensitivity to ASB triggering four different specimen geometries with different stress concentrators were tested and analyzed. After identification of all material constants for Ti-6Al-4V alloy in the advanced constitutive relation, this relation has been introduced into the finite element code ABAQUS Explicit. The constitutive relation accounts for the effects of strain hardening, rate sensitivity and thermal softening. In this way a complete dynamic approach with elastic-plastic wave propagation, adiabatic shear band development and failure could be studied numerically for all four geometries with different stress concentrators. In addition, the Critical Impact Velocity (CIV) in shear has been estimated analytically and numerically for Ti-6Al-4V. A very good agreement between experiments and numerical simulations are achieved.

Impact tension of sheet metals - Effect of initial specimen length

It is well known that a Specimen for impact testing of materials must be optimized concerning its dimensions. The main reason is to reduce strain gradients due to the effects of elastic-plastic wave propagation. On the other hand, when Split Hopkinson Bar (SHB) is applied for tension test, the net displacement of the specimen ends is very limited, usually from 2.0 to 3.0 mm. Thus, to reach maximum strain 0.5 the specimen length must be reduced to dimensions from 4.0 mm to 6.0 mm. Consequently small diameter, or lateral dimensions in case of flat specimen, must be applied to assure one-dimensional deformation. Such small lengths substantially perturb the real material behavior to be determined. So the main motivation of this study was to perform a systematic analysis, numerical and analytical, to find differences in behavior of short and long specimens loaded in impact tension. The FE code Abaqus/Explicit has been used to simulate several specimen lengths from 10 mm to 40 mm, and several velocities from 10 m/s to 100 m/s.

High-order simulation of the elastic–plastic wave equation in two space dimensions

In this paper, we present a numerical scheme for the simulation of elastic–plastic waves in solids in one and two space dimensions, i.e. for the two cases of so-called anti-plane shear and plane strain. In principle, the method presented can be used for simulations of any order in space and time and for any hysteresis model that allows the description of plastic flow in the form of an ODE. Furthermore, we show numerical results in one and two space dimensions.