Dynamic plasticity of beryllium in the inertial fuel fusion capsule regime
Damian C. Swift, Thomas E. Tierney, Sheng-Nian Luo, Roberta N. Mulford, George A. Kyrala, Randall P. Johnson, James A. Cobble, Daviid L. Tubbs, Nelson M. Hoffman
aa r X i v : . [ c ond - m a t . m t r l - s c i ] N ov Dynamic plasticity of berylliumin the inertial fusion fuel capsule regime
Damian C. Swift ∗ ,Thomas E. Tierney, Sheng-Nian Luo,Roberta N. Mulford, George A. Kyrala, Randall P. Johnson,James A. Cobble, David L. Tubbs, Nelson M. Hoffman Los Alamos National Laboratory, Los Alamos, NM 87545, USA Abstract
The plastic response of beryllium was investigated on nanosecond time scales ap-propriate for inertial confinement fusion fuel capsules, using laser-induced shockwaves, with the response probed with surface velocimetry and in-situ x-ray diffrac-tion. Results from loading by thermal x-rays (hohlraum) were consistent with moreextensive studies using laser ablation. Strong elastic waves were observed, up to ∼ Preprint submitted to Elsevier Science LA-UR-05-4739 – May 31, 2005 ey words: inertial confinement fusion, dynamic plasticity, shock, beryllium,ablation, x-ray diffraction, mixture models, spall
The dynamic plasticity of beryllium on nanosecond time scales is importantin the development of inertial confinement fusion (ICF) (21; 2; 5). The ini-tial experiments on thermonuclear ignition at the National Ignition Facilitywill use 192 laser beams to deliver 1-2 MJ of energy inside a hohlraum. Theresulting flux of soft x-rays will implode a spherical capsule containing cryo-genic deuterium-tritium (DT) fuel. If the implosion is sufficiently vigorous anduniform then thermonuclear burn is predicted to propagate from the centralhotspot. The implosion has a relatively high aspect ratio (initial to final radius)of around 30, and is unstable with respect to spatial variations in thickness,surface finish, or radiation drive.A leading material for the fuel capsule is Be containing up to around 1 weightpercent of Cu to increase its opacity to the radiation drive. Compared to al-ternative capsule materials such as plastic, Be is dense and has a high thermalconductivity, which helps maintain a uniform layer of DT ice. However, Be ex-hibits elastic and plastic anisotropy when subjected respect to mechanical orthermal loading. This anisotropic response is likely to induce velocity fluctua-tions from the hohlraum drive, which may seed significant implosion instabili- ∗ Corresponding author
Email address: [email protected] (Damian C. Swift). This work was performed under the auspices of the U.S. Department of Energyunder contract W-7405-ENG-36. ∼ µ m and ∼
10 ns, has not been characterizedpreviously, so this work includes the development and calibration of appro-priate models, continuum mechanics simulations in which the microstructureis resolved in order to predict the velocity fluctuations directly, and the ex-perimental validation of predictions of anisotropic response and instabilityseeding. The continuum mechanics simulations are computationally expensivein three dimensions, so some systematic studies of the velocity fluctuationsare performed in fewer dimensions. A mixture model, originally developed forreactive flow studies (9) but subsequently used for other applications includingthe shock compression of porous minerals (6), was generalized to predict theaverage response of strong, polycrystalline materials, as an inline treatmentof texture at coarser resolutions.This paper discusses experiments performed to investigate the elastic-plasticresponse of Be on nanosecond time scales, and the development of physically-based models to describe the behavior. The mixture model, with extensionsfor strength, is presented. The treatment of strength should allow materialdamage and failure to be modeled seamlessly; damage pertinent to dynamicfailure by spallation is discussed.
Plastic flow in Be under dynamic loading was investigated experimentallyusing laser ablation to induce shock waves. Experiments were performed at theTRIDENT laser facility at Los Alamos, which has been used to induce shocks3nd quasi-isentropic compression waves by ablation of a free surface (17; 15),and also to induce loading and accelerate flyer plates by ablation through atransparent substrate to act as confinement (16). TRIDENT has a Nd:glasslaser system with a fundamental wavelength of 1054 nm. For more efficientcoupling with matter, the pulse was frequency-doubled to 527 nm (green).Operating in nanosecond mode, the pulse comprised up to 13 elements 180 pslong for each of which the intensity could be set independently, allowing theirradiance history to be controlled, with a total energy of up to 250 J. In eachexperiment, the planar sample was irradiated on one side, using a Fresnel zoneplate was distribute the laser energy uniformly over a spot 5 mm in diameter.(Fig. 2.)Velocity histories were obtained at the surface opposite the laser ablation,by laser Doppler velocimetry over a line 600 µ m long centered opposite thecenter of the ablation spot. In some experiments, a second laser beam wasused to generate x-rays by focusing it over a ∼ µ m spot on a Ti foil, 10 µ mthick; the plasma resulting from a pulse energy of 100-200 J in 1-2.5 ns cooledby emitting predominantly He- α -like line radiation (Fig. 3). The x-rays wereused to obtain diffraction measurements in situ during the shock wave eventand thus to monitor the response of the crystal lattice directly (12).To simplify the analysis and interpretation of shock wave experiments, it isdesirable that the loading history should be constant for some period, thenfall rapidly to zero. A disadvantage in using ablative loading compared withflyer impact experiments is that the relation between irradiance history andpressure history applied to the sample is not simple: for instance, a constantirradiance does not lead to a constant drive pressure (17). The irradiance his-tory must in general be tailored to the material, desired pressure, and pulse4uration. Often on nanosecond time scales, a constant pressure requires anirradiance history which increases by several tens of percent over the durationof the pulse. Tailored pulse shapes such as this can be generated at TRI-DENT because of the flexibility in pulse construction, but would generallyrequire several iterations because of inaccuracies in radiation hydrodynamicsimulations used to predict the desired shape, making it necessary to per-form experiments on shock flatness to verify and adjust the shape. The pulseshape and energy produced by the laser were not perfectly reproducible, sothere is a limit to the accuracy with which the pressure history can be spec-ified – e.g., constant – on a given shot. Rather than attempting to use anideal irradiance history, the pulse shape was measured on each shot and usedin radiation hydrodynamics simulations to predict the pressure history at apoint close to the ablated surface. The pressure history was then used as atime-dependent boundary condition in continuum mechanics simulations, al-lowing more detailed models of the behavior of condensed matter to be usedwithout requiring the simulation program to include radiation physics also.This procedure has been found to work well for a range of applications onmetallic and non-metallic elements, and intermetallic compounds (17; 14; 15),largely because relevant material properties in the plasma state – the equationof state (EOS), opacity, and conductivities – can be predicted with adequateaccuracy. For the pressures of a few tens of gigapascals considered here, theradiation hydrodynamics simulations predicted that less than a micrometerof material was ablated. Recovered samples have shown only a thin ( ∼ µ m)layer of the remaining material significantly affected by the heat associatedwith the ablation process.At a few tens of micrometers, the sample thickness was two orders of magni-5ude less than the spot size, so the experiments were one-dimensional duringthe passage of the initial shock and subsequent release from the drive surfaceand the surface monitored by velocimetry. One-dimensional simulations ofradiation hydrodynamics and continuum mechanics were adequate. The radi-ation hydrodynamics simulations used the computer programs ‘LASNEX’ (22)and ‘HYADES’ (4). The continuum mechanics simulations used the program‘LAGC1D’ (20). These programs used second-order finite difference represen-tations of space and time with Lagrangian (material-following) derivatives,and artificial viscosity to stabilize shocks. The plasticity experiments on Be included single crystals and rolled foils. Thecrystals were cut nominally parallel to (0001) planes from a boule grown byzone refining, and were polished to 30-40 µ m thick. The rocking curves werearound 2 ◦ wide, full width, half maximum (Fig. 4). The rocking curve wasdetermined statically by diffraction of Cu K α x-rays, integrated over almostthe whole of each sample. The profile of the dynamic diffraction lines wasconsistent with the static rocking curve, though there was some sign that inthe dynamic diffraction geometry, where the source was quite close to thesample, the width of the diffracted lines was slightly narrower. The foils werecommercial high-purity material (at least 99.8% Be). The texture, measured byx-ray orientation mapping, was found to be with c -axes oriented predominantlynormal to the foil – i.e., similar to the single crystals but with a distributionof angles ∼ ◦ wide (full width) – and the a -axes distributed continuously inthe plane, though clearly exhibiting preferential directions (Fig. 5). Compared6ith the single crystals, the foils presumably had been subjected to a muchgreater degree of plastic work.In the velocity history records from the single crystal experiments, the elas-tic precursor was very clear with the lowest pressure drive (mean pressure15 . ∼ Several models have been developed to describe the plastic response of Beunder dynamic loading (11), based on gas gun experiments using samplesof the order of millimeters thick. The laser-driven samples were a few tensof micrometers thick, so any time-dependence in the plastic flow processeswould be expected to lead to a higher flow stress and thus a higher velocity inthe elastic precursor; this is in fact what was observed. One of the publishedmodels (Steinberg-Lund) is time-dependent, but these experiments exploredtime scales far shorter than the regime in which the model was calibrated, soone would not expect it necessarily to perform well.The speed of the free surface from the elastic precursor was a significant frac-tion of the speed induced by the plastic shock. The elastic wave reaches thesurface first, reflects from it, and rattles between the surface and the shock.For a given shock pressure, the maximum speed seen at the free surface de-pends on the amplitude of the elastic wave. For a perfect model of the drivepressure, one would not expect the peak free surface velocity to match theexperiment unless the elastic wave was modeled adequately.Spall was ignored in most of the simulations, so the calculated surface velocity8ecelerated close to zero after the end of the drive pulse.
For reference, simulations were performed with no strength model, i.e. thescalar EOS only. A cubic Gr¨uneisen form was used (11); it was found previouslythat all common EOS agree for the mechanical response of Be in the regimeexplored by these experiments. With no elastic wave, the peak surface velocitywas higher than observed; the arrival time of the shock was consistent withthe experiment. (Fig. 10.)
As the plastic behavior of Be had not previously been explored in the timeand length scales investigated here, it is prudent to perform simulations usinga model with as few parameters as possible. In the elastic-perfectly plasticstrength model, the material state is described by the isotropic state and theelastic strain deviator. The constitutive model comprises the EOS, a shearmodulus G , and a flow (or yield) stress Y . The elastic response is σ = Gǫ e ; ifthe norm of the stress || σ || < Y then ˙ e p = 0; otherwise ˙ e p = ˙ e a .Parameters reported for beryllium are G = 151 GPa, and Y between 0.33 and1.3 GPa. With values in this range, the amplitude of the precursor was muchlower than observed experimentally. The initial peak could be reproducedwith Y ≃ . . The Steinberg-Guinan model (11) is a generalization of the elastic-plasticmodel to include work-hardening and thermal softening. and varying the ini-tial plastic strain ǫ p to investigate the sensitivity. No set of parameters wasfound which was capable of reproducing the detailed structure of the precur-sor. If the maximum flow stress was increased to give an approximate matchto the amplitude of the precursor, the amplitude and duration of the plasticpeak became very different to the observed velocity record.The Steinberg-Lund model (11) is a generalization of the Steinberg-Guinanmodel to include time-dependence. Using the published parameters for Be, theamplitude of the elastic wave was lower than observed and the plastic shockreleased much earlier then observed. No set of parameters was found whichreproduced the observed velocity history. Rather than attempting to test and calibrate explicit microstructural plasticitymodels directly against the experimental data, simple extensions of the elastic-perfectly plastic model were made to include time dependence with as fewother parameters as possible. Thus the systematic behavior observed in the10ingle crystal experiments could be investigated in isolation from the effectsof different terms in the more complicated models.The plastic strain rate was chosen initially to be˙ ǫ p = − αGǫ [1 − exp( − T ∗ /T )] (1)where the shear modulus G , activation temperature T ∗ , and rate multiplier α are in principle functions of the mass density ρ , but were kept constant forthe low compressions studied here. The activation temperature was initiallychosen to match the Steinberg-Lund value, T ∗ = 3600 K, and the sensitivity to α and T ∗ was explored. It was possible to reproduce the observed wave profilequalitatively; in particular, deceleration could be induced following the ini-tial elastic peak. However, it was not possible to reproduce the precursor andthe rise of the plastic wave very accurately. The simulations exhibited manysmall-amplitude spikes, caused by under-damped reverberations of the pre-cursor. Work-hardening from plastic flow behind the initial wave would causea smearing of the plastic wave and should act to damp out the reverberations,but this was not investigated in the context of this model.A slightly more realistic flow model was investigated to add the flexibility toreproduce the characteristic decay time of the precursor independently of itsinitial amplitude. In reality, the barrier for defect hopping mechanisms in acrystal is altered by the strain. The potential of an atom or defect with respectto displacement is periodic, with an amplitude like sin ( πǫ/b ) where ǫ is thelinear displacement and b the length of the Burgers vector in the directionof interest. Thermally-activated hopping may in principle occur in either the11irection of increasing or decreasing strain energy; the net rate is˙ ǫ p = α ( ρ ) { exp[ − T ∗ (1 − γ ) /T ] − exp[ − T ∗ (1 + γ ) /T ] } (2)where γ ( ǫ e ) ≡ sin πǫ e b ( ρ ) ! . (3)The reference period for elastic strain was taken to be a constant, b ( ρ ) = 1.The activation temperature was again chosen to match the Steinberg-Lundvalue, T ∗ = 3600 K, and the sensitivity to α was explored. For α ≤ α = 100/ns, the shock had startedto split into a precursor and a plastic wave, though elastic reverberations inthe plastic wave were pronounced. For α ≥ α = 2000/ns, though the acceleration during theplastic wave was sharper. As with the plastic relaxation model, overall rise timeof the plastic wave matched the experiment because of the delay introducedby the step during the rise. The release wave after the peak occurred at thesame time, and some structures in the peak resembled the experiment.For defect-induced plastic strain, the rate is˙ ǫ = R d | ~b | l where l is the length of dislocations per volume of material, | ~b | the Burgersvector (projected onto the direction of interest), and R d the jump rate of adefect. One could assume that a defect jog must comprise N atoms in orderto be stable, where N could be calculated given the line energy of the defect12ompared with the strain energy reduced by motion. If the probability rateof a single atom jumping is R a , the probability rate of a stable jog forming is R Na . As a simple estimate, the local jump rate R a occurs over a length | ~b | , so˙ ǫ = R a | ~b | l. For Be, | ~b | ≃ θ . Assuming the curvature is caused by a population of defects ofthe same orientation, the linear spacing between defects is | ~b | /θ , so the defectdensity l = θ | ~b | . For a 2 ◦ rocking curve in the Be crystals considered here, this implies l ≃ . × /m. The rate at which individual atoms attempt to jump and thusmediate motion of a defect is ∼ Hz, thus the prefactor α = 2000/ns isconsistent with the rate of deformation that one would expect.Spallation of the sample should really be included through a failure modelintimately connected with the plasticity model. Here, simple spall simulationswere performed to test whether the results would be at all consistent withthe experiment, given the strong and structured elastic precursor. Spall wasmodeled using the ‘minimum pressure’ method: the EOS was modified so thatthe minimum pressure it could produce was limited to a specified value –the spall strength, defined as a negative number – and any greater degree oftension would cause unbounded expansion, leading to a region of low density.The degree of deceleration after the plastic peak was reproduced with a spallstress ∼ .5 Microstructural plasticity The simple models discussed above are adequate to demonstrate that mi-crostructural processes explain the plastic behavior of a single orientation ofBe crystals, but a more complete model of rate-dependent crystal plasticity isnecessary for the resolved-microstructure simulations of instability seeding forICF; the discussion below follows the spirit of Rashid and Nemat-Nasser (10).The crystal lattice is described locally by the lattice vectors, { a, b, c } ; thesecan be defined with respect to standard conventions of orientation ( a parallelto the x -axis, b in the xy plane) by a local rotation matrix, R . The EOS andelasticity give the stress tensor s ( a, b, c ). Plastic flow is mediated by defects– dislocations or disclinations – which act along a set of slip systems definedwith respect to the lattice vectors.Each slip system is characterized by a plane (normal ~n ) and a direction ~d inthe plane. Given the stress tensor s , the stress resolved along this slip systemis s r = ~d.s.~n = ~d.σ.~n (4)since ~n and ~d are orthogonal. The plastic flow rate in the system is˙ ǫ r = Z ( ρ, T ) { exp [ − T ∗ (1 − γ ) /T ] − exp [ − T ∗ (1 + γ ) /T ] } f ( ρ d /ρ o ) (5)where ρ d is the dislocation density, ρ o the obstacle density, T ∗ the barrierenergy (again expressed as a temperature), and γ ( ǫ r ) ≡ sin πǫ e b ( ρ ) ! (6)14here ǫ r = ~d.e.~n = ~d.ǫ.~n. (7)Given a plastic strain rate ˙ ǫ r in a system, the strain tensor changes as˙ e = ˙ ǫ = − X r ˙ ǫ r ~d ⊗ ~n (8)(10). Note that ǫ is always symmetric, though contributions from individualslip systems may not be. So far, the strain rate in a given slip system is identicalwith the simple hopping-based model above except for O (1) geometrical termsdescribing the resolved shear stress on the system and the resolved strain, andthe explicit magnitude in terms of density of dislocation length. Further elasticprecursor measurements on Be crystals of different orientations will be neededfor a complete calibration of the crystal plasticity model.Work-hardening enters crystal plasticity through the interactions between dis-locations or twin boundaries and obstacles, including other dislocations. Thedislocation density and obstacle density increase with plastic strain:˙ ρ d = F ( ρ, ρ o ) ˙ ǫ r + F ( σ r ) , ˙ ρ o = G ( ρ, ρ o ) ˙ ǫ r (9)where F and F are dislocation generation by flow (e.g. Frank-Reade sources)and by direct nucleation respectively. These parameters will be calibratedagainst molecular dynamics simulations, and validated by comparison withthe slope and detailed shape on the main plastic wave.15 Strength of heterogeneous mixtures
A large amount of work has been done on interpreting shock wave data onheterogeneous materials – particularly porous ones – in terms of propertiesof pure materials (19). Work has also been done to predict restricted sets ofmechanical properties such as the shock Hugoniot, given the composition orinitial porosity (19; 3). However, relatively little has been done to developmodels capable of predicting wide-ranging response in a form satisfying therequirements of general continuum mechanics simulations.It is highly desirable to have a model which predicts the response of a het-erogeneous material given models for homogeneous components and a repre-sentation of the microstructure. For mixtures and materials of low porosity,models of the response for use in continuum simulations have been developedfrom experiments on the heterogeneous material itself, without including amodel of the “pure” components (1; 18; 7). This approach is pragmatic, buthas significant disadvantages. If interest in a different composition grows, fur-ther experiments are needed to calibrate a new model.
Ab initio techniques areincreasingly capable of predicting the response of pure elements and of stoi-chiometric compounds; these theoretical models cannot be applied to mixturesand porous materials without a predictive model of heterogeneous materials.Finally, for some applications, the composition or microstructure of a hetero-geneous material may vary continuously in space, or may evolve in time. Thisis the case for sputtered Be-Cu alloy in ICF, where the texture may changeradially through the capsule, particularly in the presence of an ion current.A predictive, inline mixture model is desirable here for simulations at varyingdegrees of detail from resolved microstructure to continuum-averaged proper-16ies.We have previously developed explicit mixture models for chemical explosives,allowing the shock initiation properties to be predicted as a function of compo-sition and porosity (13; 9). These models were essentially hydrodynamic, i.e.they did not incorporate elastic and plastic effects explicitly in the mixturemodel or the stress field generated by the material. Plasticity was includedas a contribution to heating in the treatment of hotspots. Here we present amore general model in which the constitutive behavior of the components isalso included.The dynamic response of a heterogeneous mixture is determined by the prop-erties of its components and the equilibration of stress and temperature be-tween them. For the propagation of a single shock wave of constant pressure,the conditions of interest may be perfect equilibration. However, there is nosuch thing as an ‘ideal’ shock wave experiment: in practice, experiments andreal applications explore shock propagation through a finite thickness of ma-terial and hence probe a finite time scale. If a steady shock is shown to exist,then the Rankine-Hugoniot equations apply and equilibrium conditions ap-ply behind the shock. In practice, the steadiness is established with a finiteaccuracy, and in general there may be equilibration processes occurring on alonger time scale which are in effect frozen during the shock wave experiment.Thus it is not always correct to assume that perfect equilibration occurs acrossa shock wave. In the case of more general loading histories, the response ofthe material may depend on its instantaneous non-equilibrium state, so thisshould be taken into account. It has been found by comparison with shockexperiments on mixtures that mixture models which ignore equilibration andstrength effects are not able to reproduce all the observed shock behavior1719). Thermally-activated processes such as chemical reactions, plastic flow,and phase transitions depend more strongly on non-equilibrium effects.The model described below was developed to act as a bridge between simu-lations with a resolved microstructure and continuum-averaged simulations.It is based on continuum-level averaging, but includes microstructural termswhich can be estimated fairly easily, or calibrated against more detailed sim-ulations. Each component is described as a pure material, i.e. with its ownEOS and strength model. The model was designed for use in continuum me-chanics, so it consists of a local state (varying with location and time insidea material subjected to dynamic loading) and a description of the materialresponse (the same throughout a particular material). Since each componenthas its own state, a stress and temperature can be calculated for each. Thematerial response model uses the local state to calculate ‘external’ propertiessuch as mass density, stress and temperature, and to calculate the evolutionof the state given applied loading and heating conditions. Stress and tempera-ture are equilibrated explicitly according to a separate time constant for each.This produces an exponential approach to mechanical and thermal equilibriumseparately. Ideal equilibration can be enforced by setting the time constantsto a small value. Stress equilibration was performed by adjusting the strainstate of each component towards equilibrium. In the absence of constitutiveeffects, stress equilibration becomes pressure equilibration; this was performedby adjusting the volume fraction of each component towards equilibrium, andexpanding or compressing along an isentrope. Temperature equilibration wasperformed by transferring heat energy at constant volume. Equilibration basedon finite rates was found to be convenient because it can be designed to useessentially the same material properties as a continuum simulation of a ‘pure’18aterial, thus is was possible to implement the heterogeneous mixture modelas a superset of existing types of material property (EOS etc) without havingto add additional types of calculation (e.g. different derivatives of the EOS).The numerical schemes are described in greater detail below. The time scalesfor equilibration may be estimated from the grain size and properties of eachcomponent. In principle, they could be considered as additional continuumvariables and evolved according to other microstructural processes. (Fig. 13.)Porous materials were represented by starting with a non-zero volume fractionof gas (air or reaction products).
We have previously reported results from reactive flow models using mixturesof EOS (13; 8; 9). These models treated the mixture as a set of discretecomponents i of volume fraction f i , each with its own state s i . This approachcan be extended in a straightforward way to equilibrate stresses rather thanpressures. The representation of the microstructure was contained entirely inthe volume fractions. Here we describe an incremental improvement describingthe microstructure with slightly more detail.For each component, a ‘fractional overlap area’ ω i was also defined; this tensorquantity expresses the connectivity of the component across the mixture ineach direction, in the same sense as stress and strain tensors. The overlapareas were used in equilibrating stress, allowing a strong component deemedto be continuous across the mixture to support a stress in the presence ofanother component – such as a void – which could accommodate strain with19 smaller stress (Fig. 14). An overlap area of zero for a component in someorientation implies that the component is continuous across the mixture inthat orientation. An overlap area of unity implies that the component stressshould be equilibrated ideally in that orientation. Intermediate values wereused to simulate granular mixtures.Each component in general has different material properties. In addition tothe properties pertaining to each ‘pure’ component, each component was asso-ciated with a tensor evolution parameter dω/dǫ describing the rate of changeof the overlap area with plastic strain normal to the overlap orientation. Thisparameter was used typically to simulate the homogenization of the mixture asshear strain breaks the continuity of components across the mixture. The evo-lution of the overlap areas was approximated by a material constant (vector) dω/dǫ where ǫ is interpreted as the strain component normal to the compo-nent of ω . A stress tensor and temperature is calculated for each component,and for the mixture as a whole. Thermal equilibration acts to reduce differences between the temperature T i ineach component, which is a function of the instantaneous state s i in that com-ponent. Given the component temperatures, we define a mean temperaturefor the mixture as a whole, weighting by the heat capacity of each component:¯ T ≡ w i T i P i w i : w i ≡ f i c v ( s i ) ρ ( s i ) , (10)where f i is the volume fraction of the i th component, c v is its specific heatcapacity, and ρ its mass density. Given the thermal relaxation time scale τ T ,20he relaxation factor over a time increment ∆ t is φ r ≡ (cid:16) − e − ∆ t/τ T (cid:17) . (11)Thus the change in specific internal energy for each component can be calcu-lated for a finite time increment,∆ e i = − φ r c v ( s i ) (cid:16) T i − ¯ T (cid:17) . (12) Mechanical equilibration acts to reduce differences between the stress τ i ineach component, which is a function of the instantaneous state s i in that com-ponent, which includes the strain. Here, stress and strain are tensor quantities,expressed in Voigt (vector) notation if desired. Given the component stresses,we define a mean stress for the mixture as a whole:¯ τ ≡ f i τ i ( s i ) (13)Given the mechanical relaxation time scale τ m , the relaxation factor over atime increment ∆ t is φ r ≡ (cid:16) − e − ∆ t/τ m (cid:17) . (14)In each component, the difference in stress from the mean can be expressedas a strain increment ∆ ǫ i toward equilibrium,∆ ǫ i = φ r ω i ( τ i − ¯ τ ) [ f i S ( s i )] − (15)21here S is the stiffness matrix for the component’s material in its instanta-neous state,[ S i ] kj ≡ ∂ [ τ i ] k ∂ [ ǫ ] j . (16)The strain increment can be expressed as a strain rate,grad u = ∆ ǫ i ∆ t , (17)and the change in volume fractions is ∂f i ∂t = 13 Tr ∂ǫ i ∂t ! . (18)In the case of a material described by a scalar EOS, the equilibration relationscan be expressed in simpler form. Now equilibration acts to reduce differencesin the component pressures p i , functions of the instantaneous states s i . In eachcomponent, the difference in pressure from the mean can be expressed as avolume change ∆ f i toward equilibrium,∆ f i = 13 Tr ω i ( p i − ¯ p ) ∂p i ∂f i ! − (19)where the mean pressure ¯ p is the volume-weighted average of the componentpressures. The isotropic stiffness is ∂p i ∂f i = c i ( s i ) ρ i f i (20)where c i is the bulk sound speed. 22 Spall as extension to heterogeneous strength
The heterogeneous strength model described above is a natural framework fordeveloping a class of models for material failure, including spall. Key elementsare the inclusion of void as a component in the mixture, and the use of theoverlap tensor to represent the orientation of growing regions of void withrespect to the direction of maximum tension. The overlap tensor allows themodel to discriminate between ductile and brittle failure (Fig. 15).The formation and propagation of cracks and voids is dominated by stressconcentrations, i.e. localization phenomena caused by the heterogeneity of themicrostructure. Stress concentration is not treated adequately by the overlaptensor, so sub-scale physics is required, at least at the level of an effective flowstress for the solid microstructure under tension. This is similar to the situationwith heating induced by pore collapse in the previous work on heterogeneoushigh explosives (9), where additional shape functions were used to model theenhanced heating around a collapsing pore, i.e. representing the width of thetemperature distribution as well as the mean. The heterogeneous model shouldnot be expected to predict the explicit formation and evolution of pre-existingor incipient damage sites.The model seamlessly treats recompaction of a damaged or spalled layer, butmay exhibit unphysical healing, unless the model of texture evolution is so-phisticated enough. 23
Conclusions
Elastic-plastic data were obtained for Be (0001) crystals using laser-drivenablative loading, with diagnostics comprising laser Doppler velocimetry and in-situ x-ray diffraction. These experiments explored the response of Be todynamic loading on nanosecond scales. The flow stress was much higher thanhas been observed on microsecond scales, and exhibited relatively complicatedtemporal structure. Important aspects of the velocity history were reproducedusing a model of plastic flow which was based on microstructural processes,forming the basis of a complete model of crystal plasticity in Be. A hetero-geneous mixture model developed previously for reactive flow studies was ex-tended to treat strength with a representation of material texture, and seemsto be a promising platform to treat spall. Accurate models of plasticity andphase changes in the homogeneous components of the mixture are needed tosupport physics-based damage and spall models. Such physics-based modelsare necessary to relate the large pool of experimental data and theoreticalwork from microsecond time scales and slower to the qualitatively and quan-titatively different behavior observed in high energy density experiments.
Acknowledgments
Scientists contributing to aspects of the experiments and theory include DanThoma and Jason Cooley (group MST-6), Doran Greening (X-7), Roger Kopp(X-1), Paul Bradley and Doug Wilson (X-2), Ken McClellan and Darrin Byler(MST-8), Marcus Knudson (Sandia National Laboratories), Pedro Peraltaand Eric Loomis (Arizona State University). TRIDENT and P-24 support24taff, including Sam Letzring, Randy Johnson, Bob Gibson, Tom Hurry, FredArchuleta, Tom Ortiz, Nathan Okamoto, Bernie Carpenter, Scott Evans, andTom Sedillo, were instrumental in the laser experiments. Target fabricationand characterization were carried out by Ron Perea, Bob Day, Art Nobile, BobSpringer (MST-7), and John Bingert (MST-6). Funding and project supportwas provided by Allan Hauer, Nels Hoffman, Cris Barnes, Steve Batha (Ther-monuclear Experiments), and Aaron Koskelo (Laboratory-Directed Researchand Development project on shock propagation at the mesoscale).
References [1] Asfanasenkov, N., Bogomlov, V.M., Voskoboinikov, I.M., 1970. General-ized shock Hugoniot of condensed substances. Zh. Pikl. Mekh. Fiz. 10, p.660.[2] Bradley, P.A., Wilson, D.C., 1999. Physics of one-dimensional capsuledesigns for the National Ignition Facility. Phys. of Plasmas 6, 11, 4293-303.[3] Bushman, A.V., Kanel’, G.I., Ni, A.L., Fortov, V.E., 1993. Intense Dy-namic Loading of Condensed Matter. Taylor and Francis, London.[4] CAS, 1998. HYADES computer program, version 01.05.11. Cascade Ap-plied Sciences Inc., Golden, Colorado.[5] Lindl, J.D., 1998. Inertial confinement fusion. Springer-Verlag, New York.[6] Luo, S.-N., Swift, D.C., Mulford, R.N., Drummond, N.D., Ackland, G.J.,2004. Performance of an ab initio equation of state for magnesium oxide.J. Phys.: Cond. Matt. 16, 30, 5435-5442.[7] Meyers, A., 1994. Dynamic behavior of materials. John Wiley and Sons,New York. 258] Mulford, R.N., Swift, D.C., 2000. Reactive flow models for the desensiti-sation of high explosive. Proc. APS Topical Conference on Shock Com-pression of Condensed Matter, held Snowbird, UT, June 1999; AIP, NewYork.[9] Mulford, R.N., Swift, D.C., 2002. Mesoscale modelling of shock initiationin HMX-based explosives. Proc. APS Topical Conference on Shock Com-pression of Condensed Matter, held Atlanta, GA, 25-29 Jun 2001, AIPCP620.[10] Rashid, M.M. and Nemat-Nasser, S., 1992. A constitutive algorithm forrate-dependent crystal plasticity, Comput. Meth. in App. Mech. and Eng.94, 201-228.[11] Steinberg, D.J., 1996. Equation of State and Strength Properties of Se-lected Materials. Lawrence Livermore National Laboratory report UCRL-MA-106439 change 1.[12] Swift, D.C., Ackland, G.J., Hauer, A., and Kyrala, G.A., 2001. Firstprinciples equations of state for simulations of shock waves in silicon.Phys. Rev. B. 64, 214107.[13] Swift, D.C., Braithwaite, M., 2000. Temperature-dependent reactive flowfor non-ideal explosives. Proc. APS Topical Conference on Shock Com-pression of Condensed Matter, held Snowbird, UT, June 1999; AIP, NewYork.[14] Swift, D.C., Gammel, J.T., Clegg, S.M., 2004. Treatment of compoundsand alloys in radiation hydrodynamics simulations of ablative laser load-ing. Phys. Rev. E. 69, 056401.[15] Swift, D.C., Johnson, R.P., 2005. Quasi-isentropic compression by abla-tive laser loading: response of materials to dynamic loading on nanosecondtime scales. Phys. Rev. E. in press.2616] Swift, D.C., Niemczura, J.G., Paisley, D.L., Johnson, R.P., Luo, S.-N.,Tierney, T.E., 2005. Laser-launched flyer plates for shock physics experi-ments. Rev. Sci. Instrum., accepted.[17] Swift, D.C., Tierney, T.E., Kopp, R.A., Gammel, J.T., 2004. Shock pres-sures induced in condensed matter by laser ablation. Phys. Rev. E. 69,036406.[18] Thouvenin, J., 1965. Effect of a shock wave on a porous solid. Proc. 4thSymposium. (Int.) on Detonation 1965, pp. 258-265, U.S. GovernmentOffice of Naval Research, White Oak, Springfield, Maryland.[19] Trunin, R.F., 1998. Shock compression of condensed matter. CambridgeUniversity Press, Cambridge.[20] WSTS, 2003. LAGC1D computer program, version 5.2 (with ARIADNEmaterial properties library, version 6.2, and C++ mathematical class li-brary, version 1.2). Wessex Scientific and Technical Services Ltd, Perth,Scotland.[21] Wilson, D.C., et al, 1998. The development and advantages of berylliumcapsules for the National Ignition Facility. Phys. of Plasmas 5, 5, 1953-9.[22] Zimmermann, G.B., Kruer, W.L., 1975. Numerical simulation of laser-initiated fusion. Comments Plas. Phys. 2, 2, 51.27 ig. 1. Frame from illustrative simulation of shock propagating through resolved mi-crostructure in Be. Upper figure shows grains (colored according to the crystal orien-tation) and the computational mesh. Lower figure shows thermal colormap of meanpressure (black: zero, white: 25 GPa). The computational domain is 100 × µ m,driven by a constant 20 GPa from the lower surface. An elastic precursor (red) canbe seen running ahead of the main shock, and spatial variations are evident acrossthe width. Velocity spectra can be extracted from these simulations and used topredict the seeding of implosion instabilities. lasma blowofffrom surface drive beamilluminates targetto induce supports shockablationlaser Dopplervelocimetry samplecompressed stateshock wave Fig. 2. Schematic cross-section of laser drive experiments. The sample was 10 to50 µ m thick; the focal spot was 5 mm in diameter. (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(c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(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) X-ray streak cameraX-ray streak cameralaser beamto drive shockcrystallaser beamto generate X-rayspoint sourceof X-rays Bragg reflection(s)Laue reflection(s)
Fig. 3. Schematic of experimental configuration for transient x-ray diffraction mea-surements. The x-ray source was 10-15 mm from the sample, and the snouts of thex-ray detectors were 50 mm apart. i n t en s i t y ( a r b i t r a r y ) Bragg angle, Cu K α (degrees)12198121991220212206ideal Fig. 4. Rocking curves (distribution of plane orientation with respect to surface) fornominally (0001) beryllium crystals. Numbers are TRIDENT shot indices. ig. 5. Orientation imaging maps of rolled beryllium foil. Each circle shows the dis-tribution of orientation of a particular Bragg reflection as a stereographic projectioncentered on the normal to the foil. The color map shows the density of the reflectionat each angle, on a logarithmic scale. The plotting software used primitive planeindices, so ‘100’ refers to the hexagonal plane (10¯10) etc. f r ee s u r f a c e v e l o c i t y ( k m / s ) time (ns)precursorsplastic waves 12198 (crystal)1218212184 Fig. 6. Free surface velocity history from single crystal (TRIDENT shot 12198:40 µ m (0001), 2.7 PW/m for 1.8 ns) and foils (shot 12182: 50 µ m, 3.2 PW/m for1.8 ns; shot 12184: 50 µ m, 6.5 PW/m for 1.8 ns). Foil histories were displaced intime so the precursors occurred at a similar time to that in the crystal. i n t en s i t y deviation from initial Bragg angle (degrees)experimentfitresiduallines unshocked plasticshock elasticprecursor Fig. 7. Time-integrated x-ray diffraction data from single crystal and fit by de-convolving using measured rocking curve (TRIDENT shot 12202: 40 µ m (0001),6 PW/m for 1.8 ns). Lines were identified by inspection and by comparison withcompression predicted using radiation hydrodynamics simulations. i n t en s i t y lattice parameter (angstrom) (decayed)precursor(uniaxial)higher pressuresat early time unshockedpeak precursor(isotropic) Fig. 8. Time-integrated x-ray diffraction data from single crystal, converted to vari-ations in lattice parameter (TRIDENT shot 12202: 40 µ m (0001), 6 PW/m for1.8 ns). Width of each bar indicates uncertainty. f r ee s u r f a c e v e l o c i t y ( k m / s ) time after start of laser drive (ns) Fig. 9. Free surface velocity history from a beryllium foil, with traces taken at differ-ent points across the sample – field of view ∼ µ m – showing spatial variationsin the precursor (TRIDENT shot 12145: 13 µ m foil, 0.6 PW/m for 3.6 ns). Thisexperiment was driven at a relatively low pressure ( ∼ f r ee s u r f a c e v e l o c i t y ( k m / s ) time after start of laser pulse (ns)experimentsimulation Fig. 10. Comparison between free surface velocity history measured for ablativeloading of a single crystal cut parallel to (0001) (TRIDENT shot 12198: 40 µ m(0001), 2.7 PW/m for 1.8 ns) and a hydrodynamic simulation in which spall wasignored. f r ee s u r f a c e v e l o c i t y ( k m / s ) time after start of laser pulse (ns)experiment0.33 GPa1.3 GPa4.0 GPa6.0 GPa Fig. 11. Comparison between free surface velocity history measured for ablativeloading of a single crystal cut parallel to (0001) (TRIDENT shot 12198: 40 µ m(0001), 2.7 PW/m for 1.8 ns) and elastic-perfectly plastic simulations using differentvalues of the flow stress Y . f r ee s u r f a c e v e l o c i t y ( k m / s ) time after start of laser pulse (ns)experimentno spall1 GPa2 GPa Fig. 12. Comparison between free surface velocity history measured for ablativeloading of a single crystal cut parallel to (0001) (TRIDENT shot 12198: 40 µ m(0001), 2.7 PW/m for 1.8 ns) and simulations using the crystal relaxation model,( T ∗ = 3600 K and α = 2000/ns), with a simple spall model, for different values ofthe spall strength. hydrocode cellregions of different materialmechanical equilibration:change volume fractions, pdv workthermalequilibration:transfer internalenergy reaction:change volumefractions Fig. 13. Schematic of the heterogeneous model. lement of mixturew: horizontal = 1, vertical = 0w: 0 < horizontal < 1, vertical = 1w: horizontal = vertical = 1 Fig. 14. Schematic of the area overlap model. Outer rectangle represents a spatialdomain described by the mixture model, e.g. a hydrocode cell. Different componentsof the mixture are represented by rectangles of different color. Overlap tensor ω describes the extent to which each component spans the whole domain, in a givendirection. void opening cracks opening Fig. 15. Schematic of different types of void growth as represented by the overlapmodel, under strain in the horizontal direction: ductile (left) and brittle (right). i n t en s i t y lattice parameter (angstrom)0.200.20.40.60.811.2 -4 -2 0 2 4 6 8 10 12 i n t en s i t y deviation from initial Bragg angle (degrees)experimentfitresiduallines hermal: time scale τ T Mean temperature ¯ T ≡ w i T i P i w i : w i ≡ f i c v ( s i ) ρ ( s i ) . Relaxation factor over time increment ∆ t : φ r ≡ (cid:16) − e − ∆ t/τ T (cid:17) . Change in specific internal energy: ∆ e i = − φ r c v ( s i ) (cid:0) T i − ¯ T (cid:1) . Mechanical: time scale τ m Mean stress: ¯ τ ≡ f i τ i ( s i )Relaxation factor over time increment ∆ t : φ r ≡ (cid:16) − e − ∆ t/τ m (cid:17) . Strain increment: ∆ ǫ i = φ r ω i ( τ i − ¯ τ ) [ f i S ( s i )] − where S is stiffness matrix: [ S i ] kj ≡ ∂ [ τ i ] k /∂ [ ǫ ] j . Thus strain rate grad u = ∆ ǫ i / ∆ t, and volume fraction change ∂ f i /∂ t = Tr ( ∂ ǫ i /∂ t ))